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✓ 4 results · all verified · 3 also independently AI-judged

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Amenability Reiter Nets and Folner Conditions

1 · Prerequisites

2 · Summary

Amenability for locally compact groups can be expressed through invariant means, Reiter's condition, and Følner sets. The page develops the links between these formulations, the role of left Haar measure, and the consequences for fixed-point properties and weak containment of the trivial representation.

The arguments use nets to handle groups without countability assumptions. They also distinguish choice-dependent results from the constructions that need no choice, and state where Haar normalization and regularity enter.

For general Haar measure, locally null Borel sets can have infinite global measure. The finite-detectability and averaging lemmas distinguish these conventions. Probability densities approximate all means on continuous tests, and topological means on all global L-infinity tests; smoothing supplies the full Reiter argument without assuming semifiniteness.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Complex $L^{\infty}$ space of a locally compact group

Definition

Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ (Left Haar integral and left Haar measure). Use the complex-valued measurability convention and modulus from Complex Haar L^p spaces and compactly supported functions. For a Borel measurable $f : G \to C$ , set $∥ f ∥_{\infty, μ} : = \inf {t > 0 : μ ({x \in G : ∣ f (x) ∣ > t}) = 0},$ with infimum $+ \infty$ when the set of such $t$ is empty. Let $L^{\infty} (G, μ; C)$ be the complex measurable functions with finite $∥ f ∥_{\infty, μ}$ , identify $f \sim g$ when $f = g$ $μ$ -almost everywhere, and write $L^{\infty} (G, μ; C) : = {[f] : f \in L^{\infty} (G, μ; C)} .$ Addition and complex scalar multiplication are $[f] + [g] = [f + g]$ and $α [f] = [α f]$ , and the norm is $∥ [f] ∥_{\infty} : = ∥ f ∥_{\infty, μ}$ . This is the complex $L^{\infty}$ space used on the amenability page; its functions are equivalence classes, not chosen representatives.

Facts & Assumptions

Given: A locally compact Hausdorff group $G$ with a fixed left Haar measure $μ$ .

[F1]

A complex measurable function is measurable exactly when its real and imaginary parts are measurable, and its modulus is measurable (Complex Haar L^p spaces and compactly supported functions).

[F2]

The essential supremum is the infimum of the almost-everywhere upper bounds and does not change when a function is changed on a null set (The essential supremum of a measurable function with respect to a measure).

[F3]

Equality almost everywhere means equality outside a measurable null set; countable unions of null sets are null by countable subadditivity (Measure-null sets and almost-everywhere statements relative to a measure, Measure spaces).

[F4]

The complex modulus satisfies $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ and $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F5]

Sums and real scalar multiples of real measurable functions are measurable (Closure properties of measurable functions used by the integral).

Proof

technique · direct

1.1F1F3F5

If $f \sim f^{'}$ and $g \sim g^{'}$ , then outside the union of their two null exceptional sets, $f + g = f^{'} + g^{'}$ and $α f = α f^{'}$ for every $α \in C$ . The real and imaginary parts of these sums and scalar multiples are real linear combinations of measurable functions, so [F5] shows that they remain complex measurable; therefore the displayed operations are well-defined on classes.

2.1F1F2F3F4step 1.1algebra

The essential-supremum norm is independent of the representative by [F2] and $∣ f ∣ = ∣ g ∣$ wherever $f = g$ . Because the set of almost-everywhere bounds is upward closed, each threshold $∥ f ∥_{\infty, μ} + ε$ and $∥ g ∥_{\infty, μ} + ε$ exceeds its infimum and is itself an almost-everywhere bound. Outside the union of their null exceptional sets, [F4] gives $∣ f + g ∣ \leq ∥ f ∥_{\infty, μ} + ∥ g ∥_{\infty, μ} + 2 ε$ . Also $∥ α f ∥_{\infty, μ} = ∣ α ∣ ∥ f ∥_{\infty, μ}$ by [F4] and scaling the threshold set (and directly when $α = 0$ ). Letting $ε ↓ 0$ gives the triangle inequality and homogeneity, so these operations preserve the finite-essential-supremum classes.

3.1F2F3step 2.1∎

If $∥ [f] ∥_{\infty} = 0$ , the upward-closed set of almost-everywhere bounds contains every $1 / n > 0$ . Thus each measurable set $E_{n} = {x : ∣ f (x) ∣ > 1 / n}$ is null. Their countable union is null by [F3], and outside it $∣ f (x) ∣ \leq 1 / n$ for every $n$ , hence $f = 0$ almost everywhere and $[f] = 0$ . The essential-supremum norm is therefore definite, so $L^{\infty} (G, μ; C)$ is a complex normed vector space.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Left-invariant means on $L^{\infty}$ of a locally compact group

Definition

Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ . Write $L^{\infty} (G) : = L^{\infty} (G, μ; C)$ for the complex a.e.-class space of Complex $L^{\infty}$ space of a locally compact group. For $g \in G$ define the left translate by $L_{g} f (x) : = f (g^{- 1} x) .$ It is well defined on classes and isometric, and $L_{g_{1}} L_{g_{2}} = L_{g_{1} g_{2}}$ .

For a bounded complex-linear functional, write $∥ m ∥ : = \sup {∣ m (f) ∣ : ∥ f ∥_{\infty} \leq 1}$ for its operator norm.

A mean on $L^{\infty} (G)$ is a complex-linear functional $m : L^{\infty} (G) \to C$ such that $m (f) \geq 0$ whenever $f \geq 0$ and $m (1_{G}) = 1$ , where $1_{G}$ is the class of the constant-one function. It is left invariant if $m (L_{g} f) = m (f)$ for every $g \in G$ and $f \in L^{\infty} (G)$ . Every mean has operator norm one and satisfies $∣ m (f) ∣ \leq m (∣ f ∣) \leq ∥ f ∥_{\infty} .$ Equivalently, a mean is a positive complex-linear functional of norm one.

Facts & Assumptions

Given: A locally compact Hausdorff group $G$ with fixed left Haar measure $μ$ and the complex normed space $L^{\infty} (G)$ .

[A1]

The left Haar measure is a nonzero Borel measure and satisfies $μ (g E) = μ (E)$ for every Borel set $E \subseteq G$ and $g \in G$ (Left Haar integral and left Haar measure).

[F1]

The elements of $L^{\infty} (G)$ are complex measurable functions modulo almost-everywhere equality; its operations are well defined, its norm is the essential supremum, and it is a complex normed vector space (Complex $L^{\infty}$ space of a locally compact group).

[F2]

Left translation $x \mapsto g^{- 1} x$ is a homeomorphism of $G$ ; a continuous map has Borel preimages of Borel sets (Topological group: multiplication and inversion are continuous, Group and abelian group, A continuous map has Borel preimages of Borel sets, The Borel sigma-algebra of a topological space).

[F3]

Complex conjugation, real and imaginary parts, and the modulus have their coordinate definitions and conjugation/multiplicativity laws, including $z \overline{z} = ∣ z ∣^{2}$ , $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , and $∣ \overline{z} ∣ = ∣ z ∣$ (Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F4]

For $z = a + b i$ , $∣ Re z ∣ = ∣ a ∣ \leq \sqrt{a^{2} + b^{2}} = ∣ z ∣$ ; the inequality follows because $b^{2} \geq 0$ and squaring is monotone on the nonnegative reals (Basic properties of the absolute value, Squaring is monotone on the nonnegatives, Squares of nonzero elements are positive).

Proof

technique · direct

1.1A1F1F2constructalgebra

For $g \in G$ , the composition $x \mapsto f (g^{- 1} x)$ is Borel measurable by [F2]. If $f = f^{'}$ outside a Borel null set $N$ , then $L_{g} f = L_{g} f^{'}$ outside $g N$ , which is null by [A1]; thus $L_{g}$ is well defined on a.e. classes. For each $t > 0$ , ${x : ∣ L_{g} f (x) ∣ > t} = g {y : ∣ f (y) ∣ > t}$ , so [A1] preserves every superlevel-set measure and hence the essential-supremum norm. Direct calculation gives $L_{e} = I$ and $L_{g_{1}} L_{g_{2}} = L_{g_{1} g_{2}}$ .

1.2F1F3F4givenalgebra

Let $m$ be a positive complex-linear functional on $L^{\infty} (G)$ . If $u$ is real-valued, write $u = u_{+} - u_{-}$ with $u_{+} = (u + ∣ u ∣) / 2$ and $u_{-} = (- u + ∣ u ∣) / 2$ , so $u_{+}, u_{-} \geq 0$ and both remain in $L^{\infty} (G)$ by [F1, F3]. Positivity makes $m (u_{+})$ and $m (u_{-})$ real, so $m (u)$ is real. Thus, writing $f = Re f + i Im f$ , one has $m (\overline{f}) = \overline{m (f)}$ . If $m (f) \neq 0$ , set $α : = \overline{m (f)} / ∣ m (f) ∣$ . By [F3], $∣ α ∣ = 1$ and $m (Re (α f)) = Re (α m (f)) = ∣ m (f) ∣$ ; since $Re (α f) \leq ∣ α f ∣ = ∣ f ∣$ by [F3, F4], positivity gives $∣ m (f) ∣ \leq m (∣ f ∣)$ . This inequality is immediate as well when $m (f) = 0$ .

2.1A1F1step 1.2givenalgebra∎

The constant-one class has norm one: $μ (G) > 0$ by nonzeroness in [A1], so its superlevel set is $G$ for $0 < t < 1$ and empty for $t \geq 1$ . For every $ϵ > 0$ , the essential-supremum definition [F1] gives $∣ f ∣ \leq (∥ f ∥_{\infty} + ϵ) 1_{G}$ almost everywhere. If $m$ is positive, then $m (1_{G}) \geq 0$ and $m (∣ f ∣) \leq (∥ f ∥_{\infty} + ϵ) m (1_{G})$ ; letting $ϵ$ decrease to zero and using step 1.2 yields $∣ m (f) ∣ \leq ∥ f ∥_{\infty} m (1_{G})$ , so $m$ is bounded. Taking the supremum on the unit ball and testing at $1_{G}$ shows $∥ m ∥ = m (1_{G})$ . Consequently positivity and $m (1_{G}) = 1$ imply $∥ m ∥ = 1$ , and positivity with $∥ m ∥ = 1$ implies $m (1_{G}) = 1$ . For a mean this also gives $∣ m (f) ∣ \leq m (∣ f ∣) \leq ∥ f ∥_{\infty}$ .

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Amenable locally compact group

Definition

A locally compact Hausdorff group $G$ is amenable if there exists a left-invariant mean $m$ on $L^{\infty} (G)$ , using the fixed left Haar measure and the translation action from Left-invariant means on $L^{\infty}$ of a locally compact group. Thus $m$ is a positive complex-linear functional with $m (1_{G}) = 1$ and $m (L_{g} f) = m (f)$ for every $g \in G$ and $f \in L^{\infty} (G)$ .

This definition imposes no countability, discreteness, compactness, or unimodularity assumption. It does not depend on the normalization of Haar measure: replacing $μ$ by $c μ$ for $c > 0$ preserves exactly the same null sets, so it gives the same almost-everywhere classes and essential-supremum norm on $L^{\infty} (G)$ . The mean and its left-invariance condition are therefore unchanged.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Reiter's condition (P1)

Definition

Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , and put $P : = {f \in L^{1} (G) : f \geq 0, ∥ f ∥_{1} = 1},$ where $f \geq 0$ means that the class has a real-valued representative that is nonnegative almost everywhere. For $g \in G$ define $(L_{g} f) (x) : = f (g^{- 1} x)$ on almost-everywhere classes. For compact $Q \subseteq G$ and $f \in P$ , set $Δ_{Q} (f) : = \sup ({0} \cup {∥ L_{g} f - f ∥_{1} : g \in Q}) .$ Then $0 \leq Δ_{Q} (f) \leq 2$ , and $Δ_{\emptyset} (f) = 0$ . The group $G$ satisfies Reiter's condition (P1) if for every compact $Q \subseteq G$ and every $ε > 0$ there is $f \in P$ with $Δ_{Q} (f) \leq ε$ .

Equivalently, there is a net $(f_{i})_{i \in I}$ in $P$ with its inherited $L^{1}$ -norm topology such that for every compact $Q$ and every $ε > 0$ some $i_{0} \in I$ satisfies $Δ_{Q} (f_{i}) \leq ε$ for all $i ⪰ i_{0}$ ; this is uniform convergence to zero on compact subsets. The set $P$ is convex and $L_{g} P = P$ for every $g \in G$ . Only left translates are used, the measure $μ$ is fixed, and no compactness assumption on $G$ is made.

Facts & Assumptions

Given: A locally compact Hausdorff group $G$ with a fixed left Haar measure $μ$ .

[A1]

For each $g \in G$ , the map $T_{g} (x) = g^{- 1} x$ is Borel measurable and measure-preserving: $T_{g}^{- 1} (E) = g E$ and $μ (g E) = μ (E)$ for Borel $E \subseteq G$ . Thus it induces $L_{g} f = f \circ T_{g}$ on almost-everywhere classes (Left Haar integral and left Haar measure, A continuous map has Borel preimages of Borel sets, Measure-preserving transformations and systems).

[F1]

$L^{1} (G)$ consists of complex measurable almost-everywhere classes with $∥ f ∥_{1} = \int_{G} ∣ f ∣ d μ$ ; if $f \geq 0$ , then $\int_{G} f = ∥ f ∥_{1}$ (Complex Haar L^p spaces and compactly supported functions).

[F2]

Integrals are invariant under measure-preserving maps, and the integral is complex-linear on $L^{1} (G)$ (Integral invariance under measure-preserving maps, The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F3]

A net is a function indexed by a nonempty directed preorder; antisymmetry is not required (Directed preorders and nets).

Proof

technique · direct

1.1A1F1F2algebra

If $f \in P$ , then [A1] makes $L_{g} f$ well-defined on classes and preserves nonnegativity. By [F2], $∥ L_{g} f ∥_{1} = \int_{G} ∣ f \circ T_{g} ∣ d μ = \int_{G} ∣ f ∣ d μ = 1$ . Thus $L_{g} P \subseteq P$ ; applying the same argument to $g^{- 1}$ and using $L_{g^{- 1}} L_{g} = I$ gives equality. Also, for every $g \in G$ , $∥ L_{g} f - f ∥_{1} \leq ∥ L_{g} f ∥_{1} + ∥ f ∥_{1} = 2$ , so the supremum defining $Δ_{Q} (f)$ is finite and lies in $[0, 2]$ , including the empty-test value zero.

1.2F1F2constructalgebra

For $f, h \in P$ and $t \in [0, 1]$ , choose nonnegative real representatives. Their convex combination is nonnegative and, by [F2], $∥ t f + (1 - t) h ∥_{1} = \int_{G} (t f + (1 - t) h) d μ = t ∥ f ∥_{1} + (1 - t) ∥ h ∥_{1} = 1$ . Therefore $t f + (1 - t) h \in P$ and $P$ is convex.

2.1F3step 1.1given

If $(f_{i})_{i \in I}$ is a net satisfying the compact-uniform condition, then for any compact $Q$ and $ε > 0$ its defining eventual estimate supplies $i_{0}$ with $Δ_{Q} (f_{i}) \leq ε$ for all $i ⪰ i_{0}$ . In particular $f_{i_{0}} \in P$ is a witness to Reiter's condition.

3.1F3constructalgebra∎

Conversely, assume Reiter's condition. Let $I$ be the set of all triples $(Q, ε, f)$ with $Q$ compact, $ε > 0$ , $f \in P$ , and $Δ_{Q} (f) \leq ε$ . Order them by $(Q, ε, f) ⪯ (Q^{'}, ε^{'}, f^{'})$ exactly when $Q \subseteq Q^{'}$ and $ε^{'} \leq ε$ . It is nonempty, since the condition at the compact singleton ${e}$ and $ε = 1$ supplies a witness. It is directed: for two indices apply the condition to the compact union of their test sets, which is compact as a finite union, and the positive minimum of their tolerances, obtaining a witness that gives a common upper bound. By [F3], the third-coordinate map $i \mapsto f_{i}$ is a net. Given any compact $Q$ and $ε > 0$ , the condition supplies $i_{0} = (Q, ε, f_{0})$ ; every $i ⪰ i_{0}$ then has $Δ_{Q} (f_{i}) \leq Δ_{Q_{i}} (f_{i}) \leq ε_{i} \leq ε$ . Every index carries its own witness, so no global choice function is used.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Left Følner nets for locally compact groups

Definition

Fix a left Haar measure $μ$ on a locally compact Hausdorff group $G$ . For a Borel set $F \subseteq G$ with $0 < μ (F) < \infty$ and a compact set $Q \subseteq G$ , put $Δ_{Q} (F) : = \sup ({0} \cup {μ (g F △ F) / μ (F) : g \in Q}) .$ The value is $0$ when $Q = \emptyset$ . The group $G$ satisfies the left Følner condition if for every compact $Q \subseteq G$ and every $ε > 0$ there is such a set $F$ with $Δ_{Q} (F) \leq ε$ .

A left Følner net is a net $(F_{i})_{i \in I}$ of Borel sets with $0 < μ (F_{i}) < \infty$ such that for every compact $Q \subseteq G$ and every $ε > 0$ there is $i_{0} \in I$ for which $Δ_{Q} (F_{i}) \leq ε$ whenever $i ⪰ i_{0}$ . This is uniform convergence to zero on compact subsets. The left Følner condition holds if and only if a left Følner net exists. In the single-set condition it is equivalent to test only compact sets containing the identity. Only left translates $g F$ occur.

Facts & Assumptions

Given: A locally compact Hausdorff group $G$ with a fixed left Haar measure $μ$ .

[A1]

Left translation is a homeomorphism, carries Borel sets to Borel sets, and preserves $μ$ ; $μ$ is finite on compact sets (Left Haar integral and left Haar measure).

[F1]

A net is a function from a nonempty directed preorder; antisymmetry is not required (Directed preorders and nets).

Proof

technique · direct

1.1A1givenalgebra

For every $g \in G$ , [A1] gives $μ (g F) = μ (F)$ , so $μ (g F △ F) \leq μ (g F) + μ (F) = 2 μ (F)$ . Thus each ratio in $Δ_{Q} (F)$ is in $[0, 2]$ and the displayed supremum is a finite real; including $0$ also defines it when $Q$ is empty. If the condition has been checked for compact sets containing $e$ , then for arbitrary compact $Q$ apply it to $Q \cup {e}$ , which is compact as a finite union of compact sets; the resulting estimate restricts to $Q$ . The reverse implication is immediate.

1.2F1given

If $(F_{i})_{i \in I}$ is a left Følner net, then for any compact $Q$ and $ε > 0$ its defining uniform-convergence condition supplies an index $i_{0}$ with $Δ_{Q} (F_{i}) \leq ε$ for every $i ⪰ i_{0}$ . In particular $F_{i_{0}}$ is Borel, has finite positive measure, and satisfies the single-set Følner estimate.

2.1A1F1constructalgebra∎

Conversely, assume the single-set condition. Let $I$ be the set of all triples $(Q, ε, F)$ with $Q$ compact, $ε > 0$ , $F$ Borel, $0 < μ (F) < \infty$ , and $Δ_{Q} (F) \leq ε$ . Order these triples by $(Q, ε, F) ⪯ (Q^{'}, ε^{'}, F^{'})$ exactly when $Q \subseteq Q^{'}$ and $ε^{'} \leq ε$ . This is a directed preorder: for two indices apply the condition to the compact union of their test sets and the positive minimum of their tolerances, obtaining a witness that gives a common upper bound. By [F1], the third-coordinate map $i \mapsto F_{i}$ is a net. Given any compact $Q$ and $ε > 0$ , the condition supplies an index $i_{0} = (Q, ε, F_{0})$ ; every $i ⪰ i_{0}$ then satisfies $Δ_{Q} (F_{i}) \leq Δ_{Q_{i}} (F_{i}) \leq ε_{i} \leq ε$ . This proves uniform convergence on compact sets. The witness-indexed set contains every possible witness, so this construction uses no global choice function.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Left-uniformly continuous bounded functions (UCB)

Definition

Let $B_{b} (G; C)$ be the actual bounded complex-valued functions on a locally compact Hausdorff group $G$ , with $∥ φ ∥_{\sup} : = \sup_{y \in G} ∣ φ (y) ∣$ . For $x \in G$ set $(L_{x} φ) (y) : = φ (x^{- 1} y)$ . Define $U C B (G) : = {φ \in B_{b} (G; C) : ∥ L_{x} φ - φ ∥_{\sup} ⟶ 0 as x ⟶ e} .$ These are actual functions, not chosen representatives of equivalence classes. They are continuous and form a closed translation-invariant subspace of $B_{b} (G; C)$ . The left translation action $G \times U C B (G) \to U C B (G)$ is jointly continuous in the sup-norm topology. For any fixed left Haar measure $μ$ , the class map $φ \mapsto [φ]$ embeds $U C B (G)$ isometrically into the complex $L^{\infty} (G, μ)$ space. In particular, a UCB function that vanishes $μ$ -almost everywhere vanishes everywhere.

Facts & Assumptions

Given: A locally compact Hausdorff group $G$ with a fixed left Haar measure $μ$ .

[A1]

The group operations are continuous, each left translation is a bijection, and $μ$ is a Borel left-invariant measure (Left Haar integral and left Haar measure).

[A2]

Every nonempty open subset of $G$ has positive $μ$ -measure (Haar measure is positive on nonempty open sets and finite on compact sets).

[F1]

$L^{\infty} (G, μ; C)$ consists of Borel measurable complex functions modulo almost-everywhere equality, with the essential-supremum norm (Complex $L^{\infty}$ space of a locally compact group).

[F2]

A continuous complex-valued function on $G$ is Borel measurable (A continuous map has Borel preimages of Borel sets).

Proof

technique · direct

1.1A1givenalgebra

If $φ \in U C B (G)$ , then it is continuous at every $t \in G$ . Indeed, for $y \to t$ put $x = y t^{- 1} \to e$ ; then $φ (y) = φ (x t) = (L_{x^{- 1}} φ) (t)$ , so $∣ φ (y) - φ (t) ∣ \leq ∥ L_{x^{- 1}} φ - φ ∥_{\sup} \to 0$ . Here $x^{- 1} \to e$ by continuity of inversion.

1.2givenalgebra

The zero function belongs to $U C B (G)$ . For $φ, ψ \in U C B (G)$ and $a \in C$ , the estimates $∥ L_{x} (φ + ψ) - (φ + ψ) ∥_{\sup} \leq ∥ L_{x} φ - φ ∥_{\sup} + ∥ L_{x} ψ - ψ ∥_{\sup}$ and $∥ L_{x} (a φ) - a φ ∥_{\sup} = ∣ a ∣ ∥ L_{x} φ - φ ∥_{\sup}$ show closure under addition and scalar multiplication. Thus it is a linear subspace of the bounded functions.

1.3A1givenalgebra

Let $φ$ lie in the norm closure of $U C B (G)$ . For $x \in G$ , choose $ψ \in U C B (G)$ with $2 ∥ φ - ψ ∥_{\sup} < ε / 2$ . Then $∥ L_{x} φ - φ ∥_{\sup} \leq 2 ∥ φ - ψ ∥_{\sup} + ∥ L_{x} ψ - ψ ∥_{\sup}$ , since left translation is an isometry for the sup norm. By the defining condition for $ψ$ , a neighborhood of $e$ makes the last term $< ε / 2$ . Hence $∥ L_{x} φ - φ ∥_{\sup} < ε$ there, so $φ \in U C B (G)$ and the subspace is closed.

1.4A1givenalgebra

For $g \in G$ , the group law gives $L_{x} L_{g} = L_{g} L_{g^{- 1} x g}$ . Therefore $∥ L_{x} (L_{g} φ) - L_{g} φ ∥_{\sup} = ∥ L_{g^{- 1} x g} φ - φ ∥_{\sup} \to 0$ as $x \to e$ , because $g^{- 1} x g \to e$ . So $L_{g} φ \in U C B (G)$ and the subspace is translation-invariant.

2.1A1step 1.3algebra

Fix $(x_{0}, φ_{0}) \in G \times U C B (G)$ . For all $x \in G$ and $φ \in U C B (G)$ , $∥ L_{x} φ - L_{x_{0}} φ_{0} ∥_{\sup} \leq ∥ φ - φ_{0} ∥_{\sup} + ∥ L_{x_{0}^{- 1} x} φ_{0} - φ_{0} ∥_{\sup}$ . The first term tends to zero as $φ \to φ_{0}$ and the second as $x \to x_{0}$ by the defining condition for $φ_{0}$ . This proves joint continuity of the left action.

3.1A2F1F2step 1.1constructalgebra∎

By [F2], each $φ \in U C B (G)$ is Borel measurable and so defines a class $[φ]$ in [F1]. Put $M = ∥ φ ∥_{\sup}$ . If $0 \leq t < M$ , then some point has $∣ φ ∣ > t$ , and continuity makes ${y : ∣ φ (y) ∣ > t}$ a nonempty open set. It has positive measure by [A2], so $∥ [φ] ∥_{\infty} \geq t$ . Also $∥ [φ] ∥_{\infty} \leq M$ because $∣ φ ∣ \leq M$ everywhere. Letting $t ↑ M$ when $M > 0$ , and noting both norms are zero when $M = 0$ , gives $∥ [φ] ∥_{\infty} = ∥ φ ∥_{\sup}$ . The class map is therefore isometric and injective; in particular, an almost-everywhere zero UCB function is identically zero.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Every locally compact Hausdorff group has an open sigma-compact subgroup

Statement

Let $G$ be a locally compact Hausdorff topological group and let $K$ be a compact neighbourhood of its identity $e$ . Set $U : = K K^{- 1}$ and, for $n \in N$ , let $U^{n}$ be the set of products of $n$ elements of $U$ , with $U^{0} : = {e}$ . Then $H : = ⋃_{n \in N} U^{n}$ is an open subgroup of $G$ and a countable union of compact subsets. Here a topological space is sigma-compact when it is a countable union of compact subsets. In particular, every locally compact Hausdorff group has an open sigma-compact subgroup. No axiom of choice is used.

Facts & Assumptions

Given: A locally compact Hausdorff topological group $G$ with identity $e$ .

[F1]

Local compactness gives a compact neighbourhood of $e$ , and every neighbourhood contains an open neighbourhood (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

[F2]

Multiplication and inversion in a topological group are continuous (Topological group: multiplication and inversion are continuous).

[F3]

Finite products and continuous images of compact spaces are compact (A product of finitely many compact spaces is compact in the product topology, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F4]

The group inverse law $(x y)^{- 1} = y^{- 1} x^{- 1}$ holds (In a group $e^{- 1} = e$ , $(g^{- 1})^{- 1} = g$ and $(g h)^{- 1} = h^{- 1} g^{- 1}$ , the order of the last product being essential).

[F5]

Induction on $N$ is valid (The natural numbers $N$ (von Neumann), The principle of mathematical induction).

[F6]

Left translations in a topological group are homeomorphisms (Left and right translations and inversion in a topological group are homeomorphisms).

Proof

technique · direct

1.1F1F2F3F4

Choose a compact neighbourhood $K$ of $e$ , which exists by [F1]. Its inverse $K^{- 1}$ is compact by [F2] and [F3], so $K \times K^{- 1}$ is compact; the continuous multiplication map sends it onto $U = K K^{- 1}$ , hence $U$ is compact by [F2] and [F3]. Since $e \in K$ , we have $K \subseteq U$ , so $U$ contains an open neighbourhood of $e$ by [F1]. Finally, [F4] gives $(x y^{- 1})^{- 1} = y x^{- 1}$ , and relabeling $x, y \in K$ shows $U^{- 1} = U$ . Thus $U$ is a symmetric compact neighbourhood of $e$ .

2.1F3F4F5step 1.1

We have $e \in U^{0}$ , $U = U^{- 1}$ , and $U^{m} U^{n} = U^{m + n}$ for all $m, n \in N$ , so $H = ⋃_{n} U^{n}$ contains $e$ , is closed under products, and is closed under inverses; hence it is a subgroup. Each $U^{n}$ is compact: $U^{0} = {e}$ is compact, and if $U^{n}$ is compact, then $U^{n + 1}$ is the continuous image of the compact product $U^{n} \times U$ under multiplication, so it is compact by [F3]. Induction [F5] proves this for every $n$ , and the displayed $N$ -indexed union makes $H$ sigma-compact.

3.1F1F6step 2.1∎

By [F1], choose an open neighbourhood $V$ of $e$ contained in $K$ ; then $V \subseteq U \subseteq H$ . For every $h \in H$ , [F6] makes $h V$ open, and the subgroup property gives $h V \subseteq H$ . Since $e \in V$ , each $h \in H$ lies in $h V$ , so $H = ⋃_{h \in H} h V$ is open. The existence of $K$ follows from local compactness, completing the claim for every locally compact Hausdorff group.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-10-08Open item page →

Finite Haar mass, compact detection, and integrable pairings

Statement

Assume AC (The Axiom of Choice). Let $G$ be an arbitrary locally compact Hausdorff group with its fixed left Haar measure $μ$ under Left Haar integral and left Haar measure, and let $A \subseteq G$ be Borel. The following are equivalent:

$A$ contains a Borel $B$ with $0 < μ (B) < \infty$ .
Some compact $K \subseteq G$ has $μ (A \cap K) > 0$ .
Some nonnegative $f \in L^{1} (G)$ has $\int_{A} f d μ > 0$ .

Call $A$ locally null when $μ (A \cap K) = 0$ for every compact $K$ . Thus a locally null Borel set is annihilated by every $L^{1}$ pairing, even though it need not be globally $μ$ -null. Positive global Haar measure alone does not imply these equivalent conditions; the counterexample in Remarks retains the original scaffold's obstruction. No sigma-compactness, semifiniteness or locally-null quotient convention is assumed.

Facts & Assumptions

Given: AC; an LCH group $G$ with fixed Haar measure $μ$ ; and a Borel set $A$ .

[F1]

Haar measure is outer regular on Borel sets, inner regular on opens and finite on compact sets (Left Haar integral and left Haar measure, Radon measure on an LCH space).

[F2]

Nonnegative $L^{1}$ classes have Borel representatives and finite integral; indicators have integral equal to the measure, and integrals are monotone and positively homogeneous (Complex Haar L^p spaces and compactly supported functions, Monotonicity and nonnegative homogeneity of the nonnegative integral, The nonnegative integral agrees with the simple integral on simple functions).

[F3]

Countable unions of measurable null sets are null by countable subadditivity; a nonnegative measurable function has zero integral exactly when it is zero almost everywhere (Finite and countable subadditivity of measures, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F4]

Compact subsets of a Hausdorff space are closed, hence Borel (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, The Borel sigma-algebra of a topological space).

Proof

technique · detect positive finite mass by compact intersections and by integrable threshold sets

1.1F1F4choosealgebra

Suppose (1), and put $b = μ (B) > 0$ . By [F1] choose open $O \supseteq B$ with $μ (O) < 5 b / 4 < \infty$ , then compact $K \subseteq O$ with $μ (K) > μ (O) - b / 4$ . Finite additivity inside the finite-measure $O$ gives $μ (O ∖ K) < b / 4$ , and therefore $μ (B \cap K) \geq b - μ (O ∖ K) > 0$ . Since $B \subseteq A$ , this proves (2). Conversely, under (2), $B = A \cap K$ is Borel by [F4] and has positive measure at most $μ (K) < \infty$ , proving (1). This argument uses compact inner approximation only for the open finite-measure set $O$ .

2.1F2F3step 1.1constructalgebra∎

Under (1), $f = 1_{B}$ is nonnegative in $L^{1}$ and $\int_{A} f = μ (B) > 0$ , so (3) holds. Conversely, suppose (3) and take a nonnegative finite-valued Borel representative of $f$ , modifying a null set if necessary. For $n \geq 0$ , each $E_{n} = A \cap {f > 1 / (n + 1)}$ is Borel with $μ (E_{n}) \leq (n + 1) \int_{G} f < \infty$ by [F2]. If every $E_{n}$ were null, their union $A \cap {f > 0}$ would be null by [F3], implying $\int_{A} f = 0$ , a contradiction. Thus some $E_{n}$ has positive finite measure and proves (1). The equivalence also proves that every locally null Borel $A$ has $\int_{A} f = 0$ for all nonnegative $f \in L^{1}$ ; applying this to $∣ u ∣$ gives annihilation of every complex $L^{1}$ pairing.

Remarks

The original claim that every globally positive Borel set contains a finite-positive subset is false under the actual Haar convention. Under AC let $D = R$ be discrete, $G = T \times D$ , and $A = {1} \times D$ . The compact open slices have common Haar measure $c > 0$ and normalized torus measure $c λ$ . Every compact set meets finitely many slices. For countable $S \subseteq D$ , arcs around $1$ in its slices can have total measure below any prescribed positive number; open inner regularity and outer regularity give $μ ({1} \times S) = 0$ . For uncountable $S$ , every open cover has positive arc measure in each slice. Some positive reciprocal threshold is exceeded on uncountably many slices; arbitrarily large finite unions of compact subsets of those slices force the open cover to have infinite measure, and outer regularity gives $μ ({1} \times S) = \infty$ . Every subset of $A$ is Borel because it is ${1} \times S$ with $S$ clopen in $D$ . Thus $A$ is locally null and globally infinite, with no finite-positive Borel subset. This explicit obstruction is retained; the repaired equivalence gives its precise finite-detectability domain instead of changing the measure convention.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Probability-density averages and locally detectable upper essential values

Statement

Assume AC (The Axiom of Choice). Let $G$ be an arbitrary LCH group with fixed left Haar measure $μ$ , and put $P = {f \in L^{1} (G) : f \geq 0, ∥ f ∥_{1} = 1}$ . For a real global- $L^{\infty}$ class $h$ , choose a bounded Borel representative and define its locally detectable upper essential value by $β (h) : = \inf {t \in R : μ (K \cap {h > t}) = 0 for every compact K \subseteq G} .$ This is a finite real number independent of that representative, and $\sup_{f \in P} \int_{G} f h d μ = β (h) .$ For complex $h \in L^{\infty} (G)$ and $α \in C$ , the exact support-function formula is $\sup_{f \in P} Re (α \int_{G} f h d μ) = β (Re (α h)) .$ For every real bounded continuous $h$ , one has $β (h) = \sup_{x \in G} h (x)$ . All $L^{\infty}$ classes here retain the global-null convention of Complex $L^{\infty}$ space of a locally compact group; locally null functions are not identified with zero. The original global-norm formula and its naive global-essential-upper-value repair fail as explained in Remarks.

Facts & Assumptions

Given: AC; an LCH group $G$ with fixed Haar measure; and a real or complex bounded Borel representative $h$ .

[F1]

A Borel superlevel set contains finite-positive mass exactly when some compact intersection detects positive mass (Finite Haar mass, compact detection, and integrable pairings).

[F2]

Global $L^{\infty}$ classes have Borel representatives, can be clipped on a global null set to be bounded, and have the essential-supremum norm (Complex $L^{\infty}$ space of a locally compact group, The essential supremum of a measurable function with respect to a measure).

[F3]

Under AC, $C_{c} (G; C)$ is dense in $L^{1} (G)$ (Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F4]

The integral is linear on $L^{1}$ , monotone on nonnegative functions and satisfies the integral triangle inequality; a nonnegative function has zero integral exactly when it is zero a.e. (The Lebesgue integral is linear on $L^{1} (μ)$ , Monotonicity and nonnegative homogeneity of the nonnegative integral, The modulus of an integral is bounded by the integral of the modulus, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F5]

Nonempty open sets have positive Haar measure, compact sets have finite measure, and relatively compact open neighborhoods exist in an LCH space (Haar measure is positive on nonempty open sets and finite on compact sets, In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular).

[F6]

Countable unions of measurable null sets are null by countable subadditivity (Finite and countable subadditivity of measures).

Proof

technique · optimize over finite-detectable superlevels, using compactly supported approximants for the upper bound

1.1F2F5F6givenconstruct

Global a.e. changes alter every compact superlevel intersection only on a null set, so the defining thresholds and $β$ are unchanged. The admissible thresholds form an upward-closed nonempty set, bounded below because $h$ is bounded and some relatively compact nonempty open set has positive finite measure by [F5]. Thus $β$ is finite. Every $t > β$ is admissible; on each fixed compact $K$ , apply this to $t = β + 1 / n$ and [F6] to obtain $h \leq β$ almost everywhere on $K$ . Also a normalized indicator of any relatively compact nonempty open set shows $P \neq \emptyset$ .

2.1F3F4step 1.1choosealgebra

Fix $f \in P$ . By [F3] choose $u_{n} \in C_{c} (G)$ tending to $f$ in $L^{1}$ and put $v_{n} = ∣ u_{n} ∣$ . Then $v_{n} \in C_{c} (G)$ , $v_{n} \geq 0$ , and $∥ v_{n} - f ∥_{1} \leq ∥ u_{n} - f ∥_{1} \to 0$ . On the compact support of $v_{n}$ , step 1.1 gives $h \leq β$ a.e., so $\int v_{n} h \leq β \int v_{n}$ . Since $h$ is bounded, [F4] gives $∣ \int (v_{n} - f) h ∣ \leq ∥ h ∥_{\infty} ∥ v_{n} - f ∥_{1}$ and $\int v_{n} \to \int f = 1$ . Passing to the limit proves $\int f h \leq β$ .

3.1F1F4step 2.1constructalgebra

If $t < β$ , it is not admissible, so some compact $K$ gives $E = K \cap {h > t}$ of positive finite measure. By [F1], $f_{E} = μ (E)^{- 1} 1_{E} \in P$ , and [F4] makes $\int f_{E} (h - t) > 0$ because $h - t$ is strictly positive on $E$ . Thus $\int f_{E} h > t$ . Letting $t ↑ β$ and combining with step 2.1 proves the signed formula. For complex $h$ and $α$ , linearity gives $Re (α \int f h) = \int f Re (α h)$ , so the same signed formula proves the complex support-function identity.

4.1F1F5step 1.1step 3.1∎

Let $h$ be real bounded continuous and put $s = \sup_{G} h$ . Since $h \leq s$ everywhere, $β \leq s$ . For $t < s$ , the superlevel ${h > t}$ is nonempty open. Choose a relatively compact nonempty open neighborhood $V$ inside it; [F5] gives $0 < μ (V) < \infty$ , and its compact closure detects this superlevel. Hence $t$ is not admissible and $β \geq t$ . Letting $t ↑ s$ gives $β = s$ , completing all claims.

Remarks

On $Z$ with counting Haar measure, $h = - 1$ has global norm one but every probability average is $- 1$ , so the original signed formula with $∥ h ∥_{\infty}$ was false. Even replacing that norm by the global signed essential supremum is insufficient in arbitrary LCH generality. In the torus-times-uncountable-discrete example retained in Finite Haar mass, compact detection, and integrable pairings, $A = {1} \times D$ is locally null and globally infinite. Thus $h = 1_{A}$ has global norm and global upper essential value one, while the finite-detectability lemma gives $\int f h = 0$ for every $f \in P$ and $β (h) = 0$ . Both original counterexamples remain; the formula now identifies the exact support value rather than silently imposing semifiniteness or changing global-null classes.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

L1 convolution smooths bounded functions into UCB

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , and set $L_{g} f (x) : = f (g^{- 1} x)$ . For $f, b \in L^{1} (G)$ and $φ \in L^{\infty} (G)$ , define

(f * φ) (x) : = \int_{G} f (y) φ (y^{- 1} x) d μ (y) (x \in G) .

This formula defines an actual bounded continuous function independent of the representatives, with $∥ f * φ ∥_{\sup} \leq ∥ f ∥_{1} ∥ φ ∥_{\infty} .$ It belongs to $U C B (G)$ and satisfies $∥ L_{g} (f * φ) - f * φ ∥_{\sup} \leq ∥ L_{g} f - f ∥_{1} ∥ φ ∥_{\infty}$ for every $g \in G$ , as well as $L_{g} (f * φ) = (L_{g} f) * φ$ . The operation is bilinear in $f$ and $φ$ . With the extended $L^{1}$ convolution of Convolution on L1 of a locally compact group, it also satisfies $(f * b) * φ = f * (b * φ) (f, b \in L^{1} (G)) .$

Facts & Assumptions

Given: AC, an LCH group $G$ with a fixed left Haar measure $μ$ , $f, b \in L^{1} (G)$ , and $φ \in L^{\infty} (G)$ .

[F1]

The complex Haar spaces consist of Borel-measurable functions modulo almost-everywhere equality; $L^{1}$ integrability is measured by $\int ∣ f ∣$ , and the $L^{\infty}$ norm is the essential supremum (Complex Haar L^p spaces and compactly supported functions, Complex $L^{\infty}$ space of a locally compact group).

[F2]

Left Haar measure is left invariant and finite on compact sets (Left Haar integral and left Haar measure). The inversion formula is $\int u (y^{- 1}) d μ (y) = \int u (y) Δ_{G} (y^{- 1}) d μ (y)$ for nonnegative Borel $u$ ; in particular inversion sends Borel null sets to null sets (Haar change of variables under inversion).

[F3]

Continuous group operations have Borel preimages of Borel sets; for fixed $x$ , the map $y \mapsto y^{- 1} x$ is continuous (Topological group: multiplication and inversion are continuous, A continuous map has Borel preimages of Borel sets).

[F4]

The complex integral is linear and satisfies $∣ \int u d μ ∣ \leq \int ∣ u ∣ d μ$ for integrable $u$ (The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus).

[F5]

Under AC, $g \mapsto L_{g} f$ is norm-continuous in $L^{1} (G)$ for every $f \in L^{1} (G)$ (Strong continuity of left and modular right translations on L1 and L2).

[F6]

Under AC, extended convolution is a bounded bilinear operation on $L^{1} (G)$ , agrees with the compactly supported formula on $C_{c} (G)$ , and satisfies $∥ u * v ∥_{1} \leq ∥ u ∥_{1} ∥ v ∥_{1}$ ; also $C_{c} (G)$ is dense in $L^{1} (G)$ (Convolution on L1 of a locally compact group, Completeness of the complex Haar L1 and L2 spaces and density of Cc, Convolution preserves compact support and is associative).

[F7]

Fubini's theorem applies to integrable functions on sigma-finite product measure spaces (Fubini's theorem for L^1 functions on a sigma-finite product). Restrictions of Haar measure to compact sets are finite, hence sigma-finite; finite products of compact sets are compact (A product of finitely many compact spaces is compact in the product topology, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Compact support, $C_{c} (X)$ , and $C_{0} (X)$ ).

[F8]

Compact subsets of a Hausdorff space are closed, and finite Borel partitions of a compact set are measurable (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, The Borel sigma-algebra of a topological space). The defining condition for $U C B (G)$ is sup-norm continuity of the left-translation orbit, and every such function is continuous (Left-uniformly continuous bounded functions (UCB)).

[A1]

AC is used through [F5] and [F6], in the exact forms stated by their suppliers (The Axiom of Choice).

Proof

technique · direct

1.1F1F2F4

Choose Borel representatives of $f$ and $φ$ , and let $M : = ∥ φ ∥_{\infty}$ . For each $x \in G$ , the preimage of a Borel null set $N$ under $y \mapsto y^{- 1} x$ is $x N^{- 1}$ , which is null by inversion and left invariance [F2]; thus changing either representative changes the integrand only on a null set. For every $η > 0$ , the set where $∣ φ ∣ > M + η$ is null by [F1], so its pullback is null and $y \mapsto f (y) φ (y^{- 1} x)$ is integrable. The integral is therefore defined for every $x$ , independent of representatives, and [F4] gives $∣ (f * φ) (x) ∣ \leq (M + η) ∥ f ∥_{1}$ ; letting $η ↓ 0$ yields $∥ f * φ ∥_{\sup} \leq ∥ f ∥_{1} M$ .

2.1F2F3F7F8step 1.1

First let $f, b \in C_{c} (G)$ , set $K : = supp f$ , $L : = supp b$ , and $C : = K L$ , and fix $x \in G$ . These are compact sets of finite Haar measure, and $K, C$ are Borel by [F2, F7, F8]. On $K \times C$ the function $Ψ (y, z) : = b (y^{- 1} z)$ is continuous. Compactness gives, for each $δ > 0$ , a finite Borel partition $E_{1}, \dots, E_{m}$ of $K$ and points $y_{j} \in K$ such that $∣ Ψ (y, z) - Ψ (y_{j}, z) ∣ < δ$ for $y \in E_{j}$ and $z \in C$ . Replacing $Ψ$ by $\sum_{j} 1_{E_{j}} (y) b (y_{j}^{- 1} z)$ makes the kernel a finite sum of product-measurable terms, so Fubini on the finite restricted measures applies [F3, F7, F8]. The error in either iterated integral is at most $δ ∥ f ∥_{1} μ (C) M$ , and hence tends to zero with $δ$ . Thus $\int_{C} \int_{K} f (y) b (y^{- 1} z) φ (z^{- 1} x) d μ (y) d μ (z) = \int_{K} \int_{C} f (y) b (y^{- 1} z) φ (z^{- 1} x) d μ (z) d μ (y)$ . For each fixed $y$ , substituting $z = y w$ in the inner integral on the right and using left invariance gives $\int_{L} b (w) φ (w^{- 1} y^{- 1} x) d μ (w) = (b * φ) (y^{- 1} x)$ . The left iterated integral is $((f * b) * φ) (x)$ by the compactly supported convolution formula, and the right one is $f * (b * φ) (x)$ . This proves associativity for $f, b \in C_{c} (G)$ .

2.2F2F4step 1.1

Linearity of the integral gives bilinearity in $f$ and $φ$ . By substituting $y = g z$ and using left invariance, $(L_{g} f) * φ (x) = \int_{G} f (z) φ (z^{- 1} g^{- 1} x) d μ (z) = (f * φ) (g^{- 1} x) = L_{g} (f * φ) (x)$ . Applying the same formula to the difference gives $∣ L_{g} (f * φ) (x) - (f * φ) (x) ∣ \leq ∥ L_{g} f - f ∥_{1} M$ for every $x$ ; taking the supremum proves the stated defect estimate.

3.1A1F5F8step 2.2step 1.1

By [A1], the AC hypothesis of [F5] is met, so $∥ L_{g} f - f ∥_{1} \to 0$ as $g \to e$ . Step 2.2 therefore gives $∥ L_{g} (f * φ) - f * φ ∥_{\sup} \to 0$ , which is exactly membership in $U C B (G)$ by [F8]. That definition also gives continuity, and step 1.1 gives boundedness.

4.1A1F6step 1.1step 2.1∎

For general $f, b \in L^{1} (G)$ , [A1] meets the AC hypothesis of [F6], so choose sequences $f_{n}, b_{n} \in C_{c} (G)$ converging in $L^{1}$ . The convolution bound in [F6] and the smoothing bound of step 1.1 imply that both $(f_{n} * b_{n}) * φ$ and $f_{n} * (b_{n} * φ)$ converge uniformly, respectively, to $(f * b) * φ$ and $f * (b * φ)$ : each difference is bounded by $(∥ f_{n} - f ∥_{1} ∥ b_{n} ∥_{1} + ∥ f ∥_{1} ∥ b_{n} - b ∥_{1}) M$ . Since the two expressions agree for every $n$ by step 2.1, their limits agree, proving associativity for arbitrary $L^{1}$ data. Together with steps 1.1–3.1 this proves the remaining assertions.

Sources

BHV, Kazhdan's Property (T), Appendix G, §G.3, printed p. 453 (PDF p. 459), states that convolution of $f \in L^{1} (G)$ and $φ \in L^{\infty} (G)$ belongs to $U C B (G)$ . Thomas, The Banach–Tarski Paradox and Amenability, Lecture 20, PDF p. 13, UCB smoothing lemma (slide labelled 10), states the same membership claim. The actual-function formula, representative independence, norm estimate, equivariance, and associativity are proved here; the source statements do not supply these details.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

A UCB-invariant mean yields a topological invariant mean

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , and let $m$ be a left-invariant mean on the actual-function space $U C B (G)$ of Left-uniformly continuous bounded functions (UCB). Thus $m$ is a positive complex-linear functional with $m (1_{G}) = 1$ and $m (L_{g} ψ) = m (ψ)$ for every $g \in G$ and $ψ \in U C B (G)$ . Fix $f_{0} \in P$ for $P$ of Reiter's condition (P1), and define $\tilde{m} (φ) : = m (f_{0} * φ) (φ \in L^{\infty} (G)),$ where $*$ is the pointwise $L^{1}$ -to- $L^{\infty}$ smoothing of L1 convolution smooths bounded functions into UCB. Then $\tilde{m}$ is a mean on $L^{\infty} (G)$ and is topologically invariant: $\tilde{m} (f * φ) = \tilde{m} (φ) (f \in P, φ \in L^{\infty} (G)) .$

Facts & Assumptions

Given: AC, an LCH group $G$ with fixed left Haar measure $μ$ , a left-invariant mean $m$ on $U C B (G)$ , and the probability densities $P$ .

[A1]

AC is assumed in the choice-function form of the axiom (The Axiom of Choice).

[F1]

$U C B (G)$ consists of actual bounded continuous functions; its left-translation orbit is sup-norm continuous, translations preserve UCB, and its sup norm agrees with the embedded $L^{\infty}$ norm (Left-uniformly continuous bounded functions (UCB)).

[F2]

The smoothing formula $f * φ (x) = \int f (y) φ (y^{- 1} x) d μ (y)$ is an actual bounded UCB function independent of representatives, with sup bound, left-equivariance and associativity $(f * b) * φ = f * (b * φ)$ for $f, b \in L^{1}$ , $φ \in L^{\infty}$ (L1 convolution smooths bounded functions into UCB).

[F3]

$C_{c} (G)$ is dense in $L^{1} (G)$ under AC. The extended convolution is a continuous bilinear operation agreeing with the Cc formula and satisfying $∥ u * v ∥_{1} \leq ∥ u ∥_{1} ∥ v ∥_{1}$ ; the Cc convolution kernel is continuous and compactly supported (Completeness of the complex Haar L1 and L2 spaces and density of Cc, Convolution on L1 of a locally compact group, Submultiplicativity of convolution in the L1 norm, Compactly supported convolution on a group, Convolution preserves compact support and is associative).

[F4]

For Cc kernels, Fubini interchanges the compactly supported Radon integrals under AC; left Haar invariance gives $\int v (y^{- 1} x) d μ (x) = \int v d μ$ (Compactly supported kernels admit commuting radon integrals, Left Haar integral and left Haar measure).

[F5]

The positive probability approximate-identity net $(e_{U})$ lies in $C_{c} (G) \cap P$ and satisfies $∥ f * e_{U} - f ∥_{1} \to 0$ for every $f \in L^{1} (G)$ under AC (L1 group algebras have a contractively bounded approximate identity, Reiter's condition (P1)).

[F6]

The integral is linear, satisfies $∣ \int u ∣ \leq \int ∣ u ∣$ , and is monotone and positively homogeneous on nonnegative measurable functions; a nonnegative function with zero integral vanishes almost everywhere (Integrable real and complex functions, and their integrals, The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F7]

On a compact subset of a Hausdorff space, a finite open cover can be disjointified into a finite Borel partition; samples can be chosen from its finitely many nonempty cells (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, The Borel sigma-algebra of a topological space, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, Every natural-number-indexed list of nonempty sets has a choice function on its family of values).

Proof

technique · direct

1.1A1F3F7algebra

First, $m$ has norm one. For a real-valued $ψ \in U C B (G)$ , $∥ ψ ∥_{\sup} 1_{G} \pm ψ \geq 0$ , so positivity and $m (1_{G}) = 1$ imply $m (ψ) \in R$ and $∣ m (ψ) ∣ \leq ∥ ψ ∥_{\sup}$ . For complex $ψ$ , choose $α$ with $∣ α ∣ = 1$ and $α m (ψ) = ∣ m (ψ) ∣$ ; conjugation preserves UCB, so $Re (α ψ) \in U C B (G)$ and $∣ m (ψ) ∣ = m (Re (α ψ)) \leq ∥ ψ ∥_{\sup}$ by positivity. Testing at $1_{G}$ gives $∥ m ∥ = 1$ . Now fix $f, b \in P$ . By [F3] choose Cc approximants $u_{n}, v_{n}$ to $f, b$ in $L^{1}$ , and replace them by $∣ u_{n} ∣, ∣ v_{n} ∣$ ; since $f, b \geq 0$ and $∣ ∣ u_{n} ∣ - f ∣ \leq ∣ u_{n} - f ∣$ , these remain convergent nonnegative Cc approximants.

2.1A1F1F2F3F7constructstep 1.1

Fix $f \in P$ and $ψ \in U C B (G)$ . Given $ε > 0$ , choose $u \in C_{c} (G)$ with $∥ f - u ∥_{1} < ε$ and put $K = supp u$ ; then $\int_{G ∖ K} f d μ < ε$ . Since $y \mapsto L_{y} ψ$ is sup-norm continuous and $K$ is compact, a finite open cover of $K$ gives a finite Borel partition $E_{1}, \dots, E_{N}$ of $K$ and sample points $y_{j} \in K$ with $∥ L_{y} ψ - L_{y_{j}} ψ ∥_{\sup} < ε$ for $y \in E_{j}$ . Set $S = \sum_{j} (\int_{E_{j}} f d μ) L_{y_{j}} ψ$ . The pointwise smoothing formula in [F2] gives $∥ f * ψ - S ∥_{\sup} \leq ε + ε ∥ ψ ∥_{\sup}$ , from the partition error on $K$ and the tail outside $K$ . Invariance and linearity of $m$ give $m (S) = m (ψ) \int_{K} f d μ$ , hence $∣ m (S) - m (ψ) ∣ \leq ε ∥ ψ ∥_{\sup}$ . By step 1.1, $∣ m (f * ψ) - m (ψ) ∣ \leq ε (1 + 2 ∥ ψ ∥_{\sup})$ ; letting $ε ↓ 0$ proves $m (f * ψ) = m (ψ)$ . This uses finite Borel partitions and pointwise scalar integrals, with no Bochner measurability or separability assumption.

2.2F2F6step 1.1

Define $\tilde{m} (φ) = m (f_{0} * φ)$ . By [F2], $\tilde{m}$ is complex-linear and $∣ \tilde{m} (φ) ∣ \leq ∥ f_{0} * φ ∥_{\sup} \leq ∥ φ ∥_{\infty}$ by step 1.1. If $φ \geq 0$ , the pointwise smoothing formula gives $f_{0} * φ \geq 0$ , hence $\tilde{m} (φ) \geq 0$ . Also $f_{0} * 1_{G} = 1_{G}$ because $\int f_{0} = 1$ , so $\tilde{m} (1_{G}) = 1$ . Therefore $\tilde{m}$ is a mean on $L^{\infty} (G)$ .

2.3F3F4F6step 1.1

For nonnegative $u, v \in C_{c} (G)$ , the compactly supported convolution formula gives $u * v \geq 0$ . Its integral is $\int_{x} \int_{y} u (y) v (y^{- 1} x) d μ (y) d μ (x) = \int_{y} u (y) \int_{z} v (z) d μ (z) d μ (y) = (\int u) (\int v)$ by [F4] and the substitution $x = y z$ . Applying this to the approximants fixed in step 1.1 gives convolution masses tending to $(\int f) (\int b) = 1$ .

3.1F2F5step 1.1step 2.1

Let $(e_{U})$ be the net of [F5]. For any $f \in P$ and $φ \in L^{\infty} (G)$ , $e_{U} * φ \in U C B (G)$ by [F2], so step 2.1 and associativity in [F2] give $m ((f * e_{U}) * φ) = m (f * (e_{U} * φ)) = m (e_{U} * φ)$ . Also [F2] gives $∥ (f * e_{U}) * φ - f * φ ∥_{\sup} \leq ∥ f * e_{U} - f ∥_{1} ∥ φ ∥_{\infty} \to 0$ by [F5]. Since $m$ is bounded by step 1.1, $m (f * φ) = \lim_{U} m (e_{U} * φ)$ , independent of $f \in P$ . Hence $m (f * φ) = m (f_{0} * φ)$ for all $f, f_{0} \in P$ .

3.2F3F6step 1.1step 2.3

The extension bound in [F3] gives $u_{n} * v_{n} \to f * b$ in $L^{1}$ for the nonnegative Cc approximants from step 1.1: the difference is bounded by $∥ u_{n} - f ∥_{1} ∥ v_{n} ∥_{1} + ∥ f ∥_{1} ∥ v_{n} - b ∥_{1}$ . Since each $u_{n} * v_{n}$ is nonnegative, $∥ Im (f * b) ∥_{1}$ and $∥ (Re (f * b))^{-} ∥_{1}$ are bounded by $∥ u_{n} * v_{n} - f * b ∥_{1}$ and tend to zero; [F6] implies $f * b \geq 0$ almost everywhere. Continuity of integration on $L^{1}$ , also in [F6], gives $\int f * b = \lim_{n} (\int u_{n}) (\int v_{n}) = 1$ by step 2.3. Thus $f * b \in P$ .

4.1F2step 3.1step 3.2step 2.2∎

For $f \in P$ and $φ \in L^{\infty} (G)$ , associativity in [F2] gives $\tilde{m} (f * φ) = m (f_{0} * (f * φ)) = m ((f_{0} * f) * φ)$ . Step 3.2 gives $f_{0} * f \in P$ , so the kernel independence proved in step 3.1 makes this value $m (f_{0} * φ) = \tilde{m} (φ)$ . Every argument of $m$ here is a smoothed UCB function by [F2]; no value of $m$ on an arbitrary unsmoothed $L^{\infty}$ input is used. Together with step 2.2 this proves that $\tilde{m}$ is a topological invariant mean.

Remark

Step 3.2 also proves that the extended $L^{1}$ convolution preserves probability densities: for all $f, b \in P$ , one has $f * b \in P$ . The local proof uses nonnegative compactly supported approximants, compact-support Radon integration, and convergence in the $L^{1}$ convolution norm.

Sources

BHV, Kazhdan's Property (T), Appendix G, §G.3, proof of Theorem G.3.1, (i) to (ii), printed pp. 453–454, proves the identity $m (f * φ) = m (φ)$ for UCB inputs, compares probability kernels using a positive approximate identity, and defines $\tilde{m} (φ) = m (f_{0} * φ)$ . Thomas, Lecture 20, slides 11–14, gives the same construction. The local proof justifies the compact-partition approximation and uses the right-L1 estimate $∥ f * e_{U} - f ∥_{1}$ with the smoothing bound; it makes no sup-norm approximate- identity claim for arbitrary L-infinity inputs.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Probability-density approximation of continuous tests and topological means

Statement

Assume AC (The Axiom of Choice). Let $G$ be an arbitrary LCH group with fixed left Haar measure $μ$ , let $P = {f \in L^{1} (G) : f \geq 0, ∥ f ∥_{1} = 1}$ and let $M$ be the means on the global-null $L^{\infty} (G)$ of Left-invariant means on $L^{\infty}$ of a locally compact group. Then:

Every $m \in M$ , finite list of bounded continuous functions $ψ_{1}, \dots, ψ_{k}$ and $ε > 0$ admit $f \in P \cap C_{c} (G)$ with $∣ \int f ψ_{j} d μ - m ([ψ_{j}]) ∣ < ε$ for all $j$ .
If $m$ is topologically invariant, meaning $m (p * φ) = m (φ)$ for every $p \in P$ and $φ \in L^{\infty} (G)$ using the smoothing of L1 convolution smooths bounded functions into UCB, then $m$ is in the weak-star closure of $P$ : every finite list of global $L^{\infty}$ classes can be approximated simultaneously by their probability-density integrals.

Here $P$ embeds into $L^{\infty} (G)^{*}$ by integration. Full weak-star density in ALL means is false under the existing Haar/global-null conventions; the original counterexample remains in Remarks. No countability or semifiniteness of $G$ is imposed.

Facts & Assumptions

Given: AC; an LCH group $G$ with fixed left Haar measure; a mean $m$ on global $L^{\infty} (G)$ ; and finite test families.

[F1]

A mean is positive, complex-linear, unital and bounded by the $L^{\infty}$ norm (Left-invariant means on $L^{\infty}$ of a locally compact group).

[F2]

Probability averages of a real bounded continuous function have supremum equal to its pointwise supremum (Probability-density averages and locally detectable upper essential values).

[F3]

Under AC, separation separates disjoint convex sets when one is open (Separation of disjoint convex sets when one is open).

[F4]

Relatively compact open neighborhoods exist, have finite positive Haar measure when nonempty, and $C_{c} (G)$ is dense in $L^{1} (G)$ under AC (In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular, Haar measure is positive on nonempty open sets and finite on compact sets, Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F5]

Smoothing $f * φ$ is an actual bounded continuous UCB function with sup norm at most $∥ f ∥_{1} ∥ φ ∥_{\infty}$ ; extended convolution agrees with Cc convolution and is bounded in $L^{1}$ (L1 convolution smooths bounded functions into UCB, Convolution on L1 of a locally compact group).

[F6]

Left invariance, inversion and right translation give $\int F (x^{- 1}) d μ (x) = \int F (x) Δ_{G} (x^{- 1}) d μ (x)$ and $\int F (x z) d μ (x) = Δ_{G} (z^{- 1}) \int F d μ$ ; $Δ_{G}$ is a continuous positive homomorphism (Left Haar integral and left Haar measure, Haar change of variables under inversion, Right translation scales left Haar measure, The modular function is a continuous homomorphism).

[F7]

Continuous compactly supported kernels on LCH products admit commuting Radon integrals under AC (Compactly supported kernels admit commuting radon integrals).

[F8]

Convolution preserves probability densities, as proved with nonnegative Cc approximants (A UCB-invariant mean yields a topological invariant mean, Remark). The integral is linear and satisfies the triangle inequality (The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus).

[F9]

Weak-star neighborhoods test finitely many evaluations, and nets may use witness-indexed directed preorders (The weak-star topology from finite evaluations, Directed preorders and nets).

Proof

technique · approximate continuous tests by probability witnesses, then smooth those witnesses to approximate a topological mean on all global-L-infinity inputs

1.1F1F2F3F4constructalgebra

For bounded continuous $ψ_{1}, \dots, ψ_{k}$ , let $D \subseteq C^{k}$ consist of their integral vectors over $P$ , and let $v = (m (ψ_{j}))_{j}$ . The set $D$ is convex. If $v \notin \overline{D}$ , choose $r > 0$ so that the open ball $B (v, r)$ is disjoint from the nonempty closed convex set $\overline{D}$ . By [F3], after reversing its sign, a nonzero real-linear functional $ℓ$ satisfies $ℓ (c) < ℓ (u)$ for every $c \in \overline{D}$ and $u \in B (v, r)$ . Choose a unit vector $w$ with $ℓ (w) > 0$ . Testing at $u = v - r w / 2$ gives $\sup_{d \in D} ℓ (d) \leq ℓ (v) - r ℓ (w) / 2 < ℓ (v)$ . Write $h (x) = ℓ ((ψ_{j} (x))_{j})$ , a real bounded continuous function. Complex linearity and positivity make $m (Re ψ) = Re m (ψ)$ and similarly for imaginary parts, so $ℓ (v) = m (h) \leq \sup_{G} h$ by [F1]. But [F2] gives $\sup_{d \in D} ℓ (d) = \sup_{f \in P} \int f h = \sup_{G} h$ , a contradiction. Thus $v \in \overline{D}$ , proving simultaneous continuous-test approximation by a density. The empty test family has a witness given by a normalized indicator of a relatively compact nonempty open set by [F4].

1.2F4F5F6F7F8algebra

If $a = 0$ or $b = 0$ , the adjoint identity below has both sides zero; assume otherwise, so both supports are nonempty. For $a, b \in C_{c} (G)$ define $a^{♯} (u) = Δ_{G} (u^{- 1}) a (u^{- 1})$ . For bounded continuous $φ$ , the compact-kernel formula and [F7] interchange the integrals of $a (y) b (z) φ (y z)$ ; left invariance and inversion [F6] then give $\int (a * b) φ = \int b (a^{♯} * φ)$ . This also holds for any bounded Borel $φ$ . Indeed put $K = supp a$ , $L = supp b$ , $C = K L$ , and choose $ψ \in C_{c} (G)$ with $∥ ψ - φ 1_{C} ∥_{1} < η$ by [F4]. The left integral changes by at most $∥ a * b ∥_{\sup} η$ . For $z \in L$ , inversion and right translation give $\int_{K^{- 1}} ∣ φ - ψ ∣ (u^{- 1} z) d μ (u) \leq M_{K} M_{L} η$ , where $M_{K} = \sup_{K} Δ_{G} (t^{- 1})$ and $M_{L} = \sup_{L} Δ_{G} (z^{- 1})$ are finite by [F6]; here $t z \in K L = C$ for $t \in K$ . The right integral changes by at most $∥ b ∥_{1} ∥ a^{♯} ∥_{\sup} M_{K} M_{L} η$ . Letting $η ↓ 0$ proves the adjoint identity for bounded Borel tests without invoking product measurability of arbitrary Borel functions.

2.1F4F8F9step 1.1constructalgebra

These witnesses can be chosen in $P \cap C_{c} (G)$ . For a witness $b \in P$ , choose $u_{n} \in C_{c} (G)$ with $u_{n} \to b$ in $L^{1}$ by [F4]. Then $∣ u_{n} ∣ \to b$ in $L^{1}$ and $\int ∣ u_{n} ∣ \to 1$ ; for large $n$ , $b_{n} = ∣ u_{n} ∣ / \int ∣ u_{n} ∣$ lies in $P \cap C_{c} (G)$ and tends to $b$ in $L^{1}$ . The finite bounded tests preserve the desired inequalities with any initial smaller error margin. Index all such witnesses by triples $(F, η, b)$ , with finite continuous test set $F$ , tolerance $η > 0$ and $b \in P \cap C_{c} (G)$ meeting it, ordered by increasing $F$ and decreasing $η$ . Step 1.1 makes this a nonempty directed preorder. Its third-coordinate net $b_{i}$ satisfies $\int b_{i} ψ \to m (ψ)$ for every bounded continuous $ψ$ , without a global witness choice. Fix also $p \in P \cap C_{c} (G)$ .

3.1F1F5F6F8F9step 2.1step 1.2constructalgebra∎

Now suppose $m$ is topologically invariant. Set $g_{i} = p^{♯} * b_{i}$ for the Cc probability witnesses of step 2.1. By [F6], $p^{♯} \geq 0$ , $∥ p^{♯} ∥_{1} = 1$ and $(p^{♯})^{♯} = p$ , so $g_{i} \in P$ by [F8]. For a global $L^{\infty}$ class $φ$ , choose a bounded Borel representative by modifying a global null set. Step 1.2 gives $\int g_{i} φ = \int b_{i} (p * φ) \to m (p * φ) = m (φ)$ , because [F5] makes $p * φ$ bounded continuous and $m$ is topological. Thus the probability integrals converge weak-star on every class; in particular any finite family is approximated simultaneously. No locally-null identification was used. Together with steps 1.1 and 2.1 this proves both claims.

Remarks

Full weak-star density in all means was the false original scaffold claim. Under AC take $G = T \times R_{d i s c r e t e}$ and $A = {1} \times R_{d i s c r e t e}$ . The finite-detectability example shows that globally null Borel subsets of $A$ are countable, while $A$ is locally null and globally infinite. Extend the co-countable filter on the discrete factor to an ultrafilter $U$ . For a global $L^{\infty}$ class choose a bounded Borel representative and set $m (φ) = \lim_{d \to U} φ (1, d)$ . Global a.e. changes affect these values on a countable set only, so this is well-defined; compactness of bounded complex disks gives the limit, continuity of complex operations gives linearity, and the essential bound/positivity outside global null sets give positivity and norm one. It is a mean with $m (1_{A}) = 1$ . Every probability pairing annihilates $1_{A}$ by finite detectability, so the weak-star neighborhood $∣ ν (1_{A}) - 1 ∣ < 1 / 2$ misses all of $P$ . This mean is not asserted to be topologically invariant: smoothing annihilates $1_{A}$ pointwise because its pullbacks are locally null and L1 pairings annihilate locally null sets. The corrected full-density conclusion is for topological means; continuous-test density remains valid for every mean.

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-10-08Open item page →

A topological invariant mean yields norm-approximately invariant densities

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , and let $\tilde{m}$ be a topological invariant mean on $L^{\infty} (G)$ : a positive complex-linear functional with $\tilde{m} (1_{G}) = 1$ and $\tilde{m} (f * φ) = \tilde{m} (φ)$ for every $f \in P$ and $φ \in L^{\infty} (G)$ , where $P$ is the set of probability densities from Reiter's condition (P1). The product $f * φ$ is the $L^{1}$ – $L^{\infty}$ smoothing of L1 convolution smooths bounded functions into UCB; products of two $L^{1}$ classes below use the extended convolution of Convolution on L1 of a locally compact group. Then there is a net $(g_{j}) \subseteq P$ such that $∥ f * g_{j} - g_{j} ∥_{1} \to 0$ for every $f \in P$ . The convergence is uniform on norm-compact subsets of $P$ : for every norm-compact $C \subseteq P$ and every $δ > 0$ there is $j_{0}$ such that $\sup_{f \in C} ∥ f * g_{j} - g_{j} ∥_{1} < δ$ whenever $j ⪰ j_{0}$ .

Facts & Assumptions

Given: AC, an LCH group $G$ with fixed left Haar measure $μ$ , a topological invariant mean $\tilde{m}$ on complex $L^{\infty} (G)$ , and the probability densities $P$ .

[A1]

AC is assumed in the choice-function form (The Axiom of Choice).

[F1]

$P = {f \in L^{1} (G) : f \geq 0, ∥ f ∥_{1} = 1}$ is the convex set of probability densities (Reiter's condition (P1)).

[F2]

Every point in a locally compact Hausdorff space has a relatively compact open neighborhood (In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular).

[F3]

Haar measure is positive on nonempty open sets and finite on compact sets (Haar measure is positive on nonempty open sets and finite on compact sets).

[F4]

Indicators of Borel sets are simple measurable functions, their simple integral is their measure, and the nonnegative integral agrees with the simple integral (Nonnegative simple measurable functions, The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions).

[F5]

A mean on complex $L^{\infty} (G)$ is positive, complex-linear and unital (Left-invariant means on $L^{\infty}$ of a locally compact group).

[F6]

Complex $L^{\infty}$ consists of Borel almost-everywhere classes (Complex $L^{\infty}$ space of a locally compact group).

[F7]

Complex numbers have real and imaginary parts (Real and imaginary parts, complex conjugation, and modulus).

[F8]

Disjoint convex sets, one open, are strictly separated by a nonzero bounded real-linear functional (Separation of disjoint convex sets when one is open).

[F9]

There is an open subgroup $H = ⋃_{n \geq 0} U^{n}$ with compact increasing exhaustion $U^{n}$ , whose left cosets partition $G$ into clopen sigma-compact subspaces (Every locally compact Hausdorff group has an open sigma-compact subgroup).

[F10]

Haar measure is Radon under the repository convention; the Borel sigma-algebra is generated by the open sets (Radon measure on an LCH space, The Borel sigma-algebra of a topological space).

[F11]

On a sigma-finite measure space, every bounded real-linear functional on real $L^{1}$ is integration against a real $L^{\infty}$ function (On a sigma-finite measure space, every bounded linear functional on $L^{p}$ is integration against a unique $L^{q}$ function).

[F12]

The complex Haar $L^{1}$ spaces are Borel almost-everywhere classes with the stated $L^{1}$ norm (Complex Haar L^p spaces and compactly supported functions).

[F13]

Under AC, $C_{c} (X; C)$ is dense in complex $L^{1} (X, μ)$ for an LCH space with Radon measure (Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F14]

Pointwise limits and their convergence sets are measurable, and monotone convergence applies to nonnegative sequences (Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable, Monotone convergence for the integral).

[F15]

Inversion satisfies $\int F (x^{- 1}) d μ (x) = \int F (x) Δ_{G} (x^{- 1}) d μ (x)$ for nonnegative Borel $F$ (Haar change of variables under inversion).

[F16]

Right translation satisfies $\int F (x z) d μ (x) = Δ_{G} (z^{- 1}) \int F d μ$ (Right translation scales left Haar measure).

[F17]

The modular function is a continuous homomorphism (The modular function is a continuous homomorphism).

[F18]

The modular function $Δ_{G}$ is positive and is the factor appearing in the left-Haar inversion formula (Modular function of a locally compact group).

[F19]

Extended convolution is a bounded bilinear operation on $L^{1}$ with $∥ u * v ∥_{1} \leq ∥ u ∥_{1} ∥ v ∥_{1}$ (Convolution on L1 of a locally compact group).

[F20]

The extended $L^{1}$ convolution preserves probability densities: for all $f, b \in P$ , $f * b \in P$ (the explicit remark recording the earlier proof's step 3.2, A UCB-invariant mean yields a topological invariant mean).

[F21]

For $f \in L^{1}$ and $φ \in L^{\infty}$ , the pointwise smoothing $f * φ$ is bounded continuous with $∥ f * φ ∥_{\sup} \leq ∥ f ∥_{1} ∥ φ ∥_{\infty}$ (L1 convolution smooths bounded functions into UCB).

[F22]

Compactly supported continuous kernels on LCH products admit commuting Radon integrals; finite products of compact spaces are compact, and continuous images of compact sets are compact (Compactly supported kernels admit commuting radon integrals, A product of finitely many compact spaces is compact in the product topology, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Topological group: multiplication and inversion are continuous).

[F28]

Left Haar integration is left invariant and linear and satisfies the integral triangle inequality (Left Haar integral and left Haar measure, The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus).

[F23]

The weak topology is defined by bounded linear functionals, and weak convergence of a net means convergence under every such functional (The dual space X^* of a normed space and its dual norm, Weak topology on a normed space, Weak convergence of nets and sequences).

[F24]

Weak-star convergence is convergence on every element of the predual (Weak star convergence).

[F25]

For a convex subset of a normed space, weak and norm closures agree under AC (Mazur theorem: weak and norm closure agree for convex sets).

[F26]

Nets may be indexed by directed preorders, including witness-indexed finite-test and tolerance triples (Directed preorders and nets).

[F27]

Compactness is intrinsic to the subspace, and every ambient open cover of a compact subspace has a finite subcover; applying this to norm balls gives a finite $ε$ -net (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it).

Proof

technique · direct

1.1F1F2F3F4construct

Choose an open relatively compact neighborhood $V$ of $e$ . By [F2, F3], $0 < μ (V) < \infty$ , and $p_{0} : = μ (V)^{- 1} 1_{V}$ belongs to $P$ ; this also proves $P$ is nonempty.

1.2A1F5F6F7F8algebra

Let $ψ_{1}, \dots, ψ_{n}$ be bounded continuous complex functions and put $z (x) = (ψ_{1} (x), \dots, ψ_{n} (x)) \in C^{n}$ , viewed as a real vector space with its maximum norm. The vector $v = (\tilde{m} (ψ_{1}), \dots, \tilde{m} (ψ_{n}))$ lies in $C : = \overset{}{co} (z (G))$ : otherwise a ball $B (v, r)$ and $C$ are disjoint convex sets, so [F8] strictly separates them by a nonzero real-linear functional $ℓ$ . Choose a unit vector $w$ with $ℓ (w) > 0$ ; applying separation to $v + r w / 2 \in B (v, r)$ gives $ℓ (v) < \inf_{c \in C} ℓ (c) = \inf_{x \in G} ℓ (z (x))$ . Now $h (x) : = ℓ (z (x))$ is bounded continuous and real-valued, and $ℓ (v) = \tilde{m} (h)$ . For real $u$ , $u = u_{+} - u_{-}$ makes $\tilde{m} (u)$ real, so $\tilde{m} (Re ψ) = Re \tilde{m} (ψ)$ and similarly for imaginary parts; positivity and normalization give $\tilde{m} (h) \geq \inf_{G} h$ , a contradiction.

1.3A1F3F9F10chooseconstruct

We first show that every bounded complex-linear functional $Λ$ on $L^{1} (G)$ is represented by a bounded Borel function. Let $H = ⋃_{n} U^{n}$ be as in [F9], choose one representative $t_{C}$ for each clopen left coset $C = t_{C} H$ using [A1], and put $K_{C, n} = t_{C} U^{n}$ . Each $C$ is the increasing union of compact sets $K_{C, n}$ of finite Haar measure, hence its restricted Haar measure is sigma-finite. Since $C$ is clopen, it is an LCH subspace and its restricted measure is Radon: Borel subsets of $C$ are Borel in $G$ , and open subsets of $C$ are open in $G$ .

1.4A1F11F12constructalgebra

For each coset $C$ , restrict $Λ$ to complex $L^{1}$ functions supported in $C$ . On real functions its real and imaginary parts are bounded real-linear functionals of norm at most $∥ Λ ∥$ ; [F11] represents them by real $a_{C}, b_{C} \in L^{\infty} (C)$ with $∥ a_{C} ∥_{\infty}, ∥ b_{C} ∥_{\infty} \leq ∥ Λ ∥$ . Thus $ϕ_{C} : = a_{C} + i b_{C}$ represents $Λ$ on complex functions supported in $C$ and $∣ ϕ_{C} ∣ \leq 2 ∥ Λ ∥$ almost everywhere. Choose Borel representatives and set them to zero on their exceptional null sets.

1.5A1F13chooseconstruct

Put $R = 2 ∥ Λ ∥$ . For each $C, n$ , the bounded function $ϕ_{C} 1_{K_{C, n}}$ is in $L^{1} (C)$ ; by [F13] choose $q_{C, n} \in C_{c} (C)$ within $2^{- n - 2}$ in $L^{1}$ . Radially clip $q_{C, n}$ to the closed disk of radius $R$ , obtaining $h_{C, n} \in C_{c} (C)$ with $∥ h_{C, n} - ϕ_{C} 1_{K_{C, n}} ∥_{1} < 2^{- n - 1}$ : pointwise, if $∣ q_{C, n} ∣ > R$ then $∣ q_{C, n} - h_{C, n} ∣ = ∣ q_{C, n} ∣ - R \leq ∣ q_{C, n} - ϕ_{C} 1_{K_{C, n}} ∣$ , and otherwise clipping does nothing, so the triangle inequality gives the stated factor two.

1.6F15F17F18F19F22algebraF28

If $f = 0$ or $v = 0$ , the adjoint identity below has both sides zero. Otherwise their compact supports are nonempty. For $f, v \in C_{c} (G)$ and bounded Borel $φ$ , define $f^{♯} (u) : = Δ_{G} (u^{- 1}) f (u^{- 1})$ . The adjoint identity $\int_{G} (f * v) φ = \int_{G} v (f^{♯} * φ)$ holds first for bounded continuous $φ$ : apply [F22] to the compactly supported continuous kernel $(y, z) \mapsto f (y) v (z) φ (y z)$ , use left invariance to set $x = y z$ , interchange the compact Radon integrals, and use inversion [F15] in $\int_{y} f (y) φ (y z)$ .

2.1F1F2F3F4step 1.1step 1.2constructalgebra

If the finite test family is empty, take $b : = p_{0}$ from step 1.1. Otherwise, given $ε > 0$ , choose a finite convex combination $\sum_{k = 1}^{r} t_{k} z (x_{k})$ within $ε / 2$ of $v$ . Continuity of the finite family gives, for each $k$ , an open neighborhood $O_{k}$ of $x_{k}$ on which $∣ ψ_{j} (y) - ψ_{j} (x_{k}) ∣ < ε / 2$ for every $j$ ; shrink it to a relatively compact open $V_{k} \subseteq O_{k}$ . Then $p_{k} : = μ (V_{k})^{- 1} 1_{V_{k}} \in P$ by [F3], and $b : = \sum_{k} t_{k} p_{k} \in P$ by [F1] satisfies $∣ \int b ψ_{j} d μ - \tilde{m} (ψ_{j}) ∣ < ε$ for every $j$ .

2.2F14step 1.5algebraconstruct

For fixed $C, m$ and every $n \geq m$ , $K_{C, m} \subseteq K_{C, n}$ , so $\int_{K_{C, m}} ∣ h_{C, n} - ϕ_{C} ∣ d μ < 2^{- n - 1}$ . By [F14], the sum of these errors is finite almost everywhere on $K_{C, m}$ , hence $h_{C, n} \to ϕ_{C}$ almost everywhere there. Since $C = ⋃_{m} K_{C, m}$ , convergence holds almost everywhere on each coset. Glue $h_{C, n}$ over the clopen cosets to a bounded continuous $H_{n}$ on $G$ . Applying the convergence-set and pointwise-limit clauses of [F14] to the real and imaginary parts shows that the convergence set of $(H_{n})$ is Borel and its limit there is Borel; set it to zero elsewhere to obtain a bounded Borel function $ϕ$ agreeing almost everywhere with each $ϕ_{C}$ on that coset. This uses only countable unions of exceptional null sets inside each individual coset.

2.3A1F13F15F16F17F18F19F21step 1.6algebraF28

For general bounded Borel $φ$ , let $K = supp f$ , $L = supp v$ , and $C = K L$ , which is compact; the pointwise Cc convolution $f * v$ vanishes off $C$ . Since $f^{♯} \in L^{1}$ and $φ$ is bounded Borel, [F21] makes $f^{♯} * φ$ bounded continuous, so the right pairing is defined. Choose $ψ \in C_{c} (G)$ with $∥ ψ - φ 1_{C} ∥_{1} < η$ by [A1, F13]. The left pairing changes by at most $∥ f * v ∥_{\sup} η$ . For fixed $z \in L$ , inversion and right translation give $\int_{K^{- 1}} ∣ φ - ψ ∣ (u^{- 1} z) d μ (u) = \int_{K} ∣ φ - ψ ∣ (t z) Δ_{G} (t^{- 1}) d μ (t) \leq M_{K} M_{L} η$ , where $M_{K} = \sup_{t \in K} Δ_{G} (t^{- 1})$ and $M_{L} = \sup_{z \in L} Δ_{G} (z^{- 1})$ are finite by [F17]. Since $f^{♯}$ is bounded and supported in $K^{- 1}$ , the right pairing changes by at most $∥ v ∥_{1} ∥ f^{♯} ∥_{\infty} M_{K} M_{L} η$ . Letting $η ↓ 0$ proves the identity for Cc $f, v$ and arbitrary $φ$ , without assuming a Borel product function is measurable for the product sigma-algebra.

3.1F26step 1.1step 2.1construct

Index by triples $(F, ε, b)$ where $F$ is a finite set of bounded continuous functions, $ε > 0$ , $b \in P$ , and $∣ \int b ψ - \tilde{m} (ψ) ∣ < ε$ for all $ψ \in F$ ; order by inclusion of $F$ and decreasing $ε$ . Steps 1.1 and 2.1 make this a nonempty directed preorder, since a witness for the union of two finite test sets and the smaller tolerance gives a common upper bound. The third-coordinate net $(b_{i})$ therefore satisfies $\int b_{i} ψ d μ \to \tilde{m} (ψ)$ for every bounded continuous $ψ$ , without a global choice function.

3.2A1F12F13F27step 1.4step 2.2algebraF28

A compact set meets only finitely many open cosets $C$ , since these cosets form an ambient open cover and [F27] supplies a finite subcover. Hence for every $u \in C_{c} (G)$ , step 1.4 and the cosetwise agreement in step 2.2 give $Λ (u) = \int_{G} u ϕ d μ$ . By [A1, F13], $C_{c} (G)$ is dense in $L^{1} (G)$ ; both sides are bounded functionals, with the integral norm at most $2 ∥ Λ ∥$ , so equality extends to every $u \in L^{1} (G)$ . Thus the full bounded dual of $L^{1} (G)$ is represented by bounded Borel functions, without claiming that an uncountable union of null sets is null.

3.3A1F6F13F15F18F19F21step 2.3F28

For arbitrary $f, v \in L^{1} (G)$ , approximate them in $L^{1}$ by Cc sequences. The class convolution bound [F19] makes $f_{n} * v_{n} \to f * v$ in $L^{1}$ ; the inversion formula gives $∥ f_{n}^{♯} - f^{♯} ∥_{1} = ∥ f_{n} - f ∥_{1}$ , and the smoothing bound [F21] gives $f_{n}^{♯} * φ \to f^{♯} * φ$ uniformly. Passing to the limit in step 2.3 proves $\int_{G} (f * v) φ = \int_{G} v (f^{♯} * φ)$ for all $f, v \in L^{1} (G)$ and $φ \in L^{\infty} (G)$ .

4.1F1F6F15F17F18F19F20F21F24step 1.1step 3.1step 3.3

Fix $p : = p_{0} \in P$ from step 1.1 and set $g_{i} : = p^{♯} * b_{i}$ . Formula [F15] gives $∥ p^{♯} ∥_{1} = ∥ p ∥_{1} = 1$ , preserves nonnegativity, and yields $(p^{♯})^{♯} = p$ ; the earlier convolution-closure proof [F20] gives $g_{i} \in P$ . For every $φ \in L^{\infty} (G)$ , step 3.3 yields $\int g_{i} φ = \int b_{i} (p * φ) \to \tilde{m} (p * φ) = \tilde{m} (φ)$ , since $p * φ$ is bounded continuous by [F21] and $\tilde{m}$ is topologically invariant. Thus the density functionals $g_{i}$ converge weak-star to $\tilde{m}$ on all of $L^{\infty} (G)$ , although the net $b_{i}$ was only chosen to approximate on continuous tests.

5.1F1F6F15F18F19F21F23step 3.2step 3.3step 4.1

Fix $f \in P$ . Formula [F15] shows that $f^{♯} \geq 0$ and $∥ f^{♯} ∥_{1} = ∥ f ∥_{1} = 1$ , so $f^{♯} \in P$ . For every $φ \in L^{\infty} (G)$ , step 3.3 gives $\int (f * g_{i}) φ = \int g_{i} (f^{♯} * φ) \to \tilde{m} (f^{♯} * φ) = \tilde{m} (φ)$ by topological invariance, while step 4.1 gives $\int g_{i} φ \to \tilde{m} (φ)$ . Hence $\int (f * g_{i} - g_{i}) φ \to 0$ . By the full dual representation in step 3.2, every bounded functional on $L^{1} (G)$ is one of these pairings; therefore $f * g_{i} - g_{i} ⇀ 0$ weakly in $L^{1} (G)$ .

6.1A1F1F19F23F25step 1.1step 5.1construct

Let $F = {f_{1}, \dots, f_{r}} \subseteq P$ be finite and nonempty. The tuple $(f_{k} * g_{i} - g_{i})_{k = 1}^{r}$ converges weakly to zero in $E = (L^{1} (G))^{r}$ with its maximum norm: each coordinate converges weakly by step 5.1, and every bounded functional on this finite product is the sum of its coordinate restrictions. Its range over $i$ lies in the convex set $D = {(f_{k} * g - g)_{k = 1}^{r} : g \in P}$ , since the map is linear in $g$ and $P$ is convex. Thus zero is in the weak closure of $D$ ; [F25] puts it in the norm closure, so for every $ε > 0$ some $g \in P$ satisfies $\max_{k} ∥ f_{k} * g - g ∥_{1} < ε$ . The empty $F$ has any witness from step 1.1.

7.1F1F26step 6.1construct

Index by triples $(F, ε, g)$ with finite $F \subseteq P$ , $ε > 0$ , $g \in P$ , and $∥ f * g - g ∥_{1} < ε$ for all $f \in F$ , ordered by inclusion of $F$ and decreasing tolerance. Step 6.1 makes this a nonempty directed preorder, so its third-coordinate net $(g_{j})$ lies in $P$ and satisfies $∥ f * g_{j} - g_{j} ∥_{1} \to 0$ for each $f \in P$ , without a global choice function.

8.1F1F19F27step 7.1∎

Let $C \subseteq P$ be norm-compact and $δ > 0$ . If $C = \emptyset$ the estimate is vacuous; otherwise choose a finite $δ / 3$ -net $f_{1}, \dots, f_{r} \in C$ and take an index after these tests with tolerance $δ / 3$ . For every later $j$ , choose $f_{k}$ with $∥ f - f_{k} ∥_{1} < δ / 3$ ; since $g_{j} \in P$ , [F19] gives $∥ f * g_{j} - g_{j} ∥_{1} \leq ∥ (f - f_{k}) * g_{j} ∥_{1} + ∥ f_{k} * g_{j} - g_{j} ∥_{1} < 2 δ / 3 < δ$ . This proves uniform convergence on $C$ and completes the lemma.

Sources

BHV, Kazhdan's Property (T), Appendix G.3, Theorem G.3.1 (ii) to (iii), printed pp. 454–455, and Thomas, Lecture 20, slides 15–17, present the weak-star-density, product-space convexity, and Mazur route. Their proof strategies are followed after the density step, but the asserted density of $L^{1}$ probabilities in the set of all $L^{\infty}$ means is not used: that separate assigned claim is refuted under the repository's Haar convention. This proof instead establishes finite-test density against bounded continuous functions, smooths by one fixed probability density to obtain weak-star convergence on all $L^{\infty}$ tests, and proves the full $L^{1}$ dual representation locally over sigma-compact cosets.

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-10-08Open item page →

An invariant mean produces a Reiter net

Statement

Assume AC. Suppose $G$ admits a left-invariant mean on $U C B (G)$ ; this holds in particular when $G$ is amenable in the sense of Amenable locally compact group. Then $G$ satisfies Reiter's condition (P1) (Reiter's condition (P1)). Consequently every amenable locally compact group satisfies (P1).

Facts & Assumptions

Given: AC, an LCH group $G$ with fixed left Haar measure $μ$ , and a left-invariant mean on actual bounded uniformly continuous functions $U C B (G)$ .

[A1]

AC is assumed in the choice-function form (The Axiom of Choice).

[F1]

$P = {f \in L^{1} (G) : f \geq 0, ∥ f ∥_{1} = 1}$ is convex and $L_{x} P = P$ for every $x \in G$ ; (P1) requires a density with the compact-test defect $Δ_{Q} (f) \leq ε$ (Reiter's condition (P1)).

[F2]

$U C B (G)$ consists of actual bounded continuous functions, is translation invariant, and its class map into complex $L^{\infty} (G)$ is an isometric embedding (Left-uniformly continuous bounded functions (UCB)).

[F3]

Amenability supplies a positive complex-linear unital left-invariant mean on complex $L^{\infty} (G)$ (Amenable locally compact group).

[F4]

Under AC, a left-invariant mean on $U C B (G)$ yields a topological invariant mean on $L^{\infty} (G)$ (A UCB-invariant mean yields a topological invariant mean).

[F5]

A topological invariant mean on $L^{\infty} (G)$ yields a net $(g_{j}) \subseteq P$ whose defects $∥ h * g_{j} - g_{j} ∥_{1}$ tend to zero uniformly for $h$ in every norm-compact subset of $P$ (A topological invariant mean yields norm-approximately invariant densities).

[F6]

For each $f \in L^{1} (G)$ , the orbit map $x \mapsto L_{x} f$ is norm-continuous (Strong continuity of left and modular right translations on L1 and L2).

[F7]

Compactness is intrinsic to the subspace, and every ambient open cover of a compact subspace has a finite subcover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it).

[F8]

A continuous image of a compact space is compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F9]

Extended $L^{1}$ convolution is bilinear and satisfies $∥ f * g ∥_{1} \leq ∥ f ∥_{1} ∥ g ∥_{1}$ ; it agrees with the compact-support convolution on $C_{c} (G)$ (Convolution on L1 of a locally compact group).

[F10]

Extended convolution preserves probability densities: $f * g \in P$ for $f, g \in P$ (A UCB-invariant mean yields a topological invariant mean, Remark).

[F11]

Under AC, $C_{c} (G)$ is dense in $L^{1} (G)$ for a Radon Haar measure (Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F12]

Left Haar measure is left invariant and Radon under the repository convention (Left Haar integral and left Haar measure).

[F13]

For $u, v \in C_{c} (G)$ , $(u * v) (y) = \int_{G} u (z) v (z^{- 1} y) d μ (z)$ (Compactly supported convolution on a group).

[F14]

Left translation preserves $C_{c} (G)$ (Translations preserve compactly supported continuous functions).

Proof

technique · direct

1.1A1F4

By [F4], the given mean on $U C B (G)$ yields a topological invariant mean $\tilde{m}$ on $L^{\infty} (G)$ .

1.2A1F9F11F12F13F14

We first prove left-equivariance of the extended convolution. For $u, v \in C_{c} (G)$ , [F13] and left invariance give, for every $x, y \in G$ , $((L_{x} u) * v) (y) = \int_{G} u (x^{- 1} z) v (z^{- 1} y) d μ (z) = \int_{G} u (w) v (w^{- 1} x^{- 1} y) d μ (w) = (L_{x} (u * v)) (y)$ , where $z = x w$ . Thus $L_{x} (u * v) = (L_{x} u) * v$ in $L^{1} (G)$ . For arbitrary $f, v \in L^{1} (G)$ choose $u_{n}, v_{n} \in C_{c} (G)$ with $u_{n} \to f$ and $v_{n} \to v$ in $L^{1}$ , using [F11]. For fixed $x$ , $L_{x} u_{n} \in C_{c} (G)$ by [F14] and $∥ L_{x} u_{n} - L_{x} f ∥_{1} = ∥ u_{n} - f ∥_{1}$ by [F12]. The convolution bound [F9] then gives $u_{n} * v_{n} \to f * v$ and $(L_{x} u_{n}) * v_{n} \to (L_{x} f) * v$ in $L^{1}$ ; passing the compact-support identity to these limits proves $L_{x} (f * v) = (L_{x} f) * v$ .

2.1A1F5step 1.1

Apply [F5] to $\tilde{m}$ from step 1.1 and fix the resulting net $(g_{j}) \subseteq P$ , with defects converging uniformly on norm-compact subsets of $P$ .

3.1A1F1F6F7F8step 2.1construct

Let $Q \subseteq G$ be compact, $ε > 0$ , and put $Q_{0} : = Q \cup {e}$ . This is compact: for an ambient open cover of $Q_{0}$ , [F7] supplies finitely many members covering $Q$ , and one additional member covers $e$ ; the ambient criterion in [F7] then gives compactness of $Q_{0}$ . The net in step 2.1 shows $P$ is nonempty, so fix $f \in P$ . By [F1], $C : = {L_{x} f : x \in Q_{0}} \subseteq P$ ; by [F6] the orbit map is continuous, and hence $C$ is norm-compact by [F8].

4.1F1F5F10step 1.2step 2.1step 3.1

By [F5] applied to the compact set $C$ , choose an index $j$ such that $∥ h * g_{j} - g_{j} ∥_{1} < ε / 2$ for every $h \in C$ . In particular, $∥ f * g_{j} - g_{j} ∥_{1} < ε / 2$ , since $f = L_{e} f \in C$ . Let $g : = f * g_{j} \in P$ by [F10]. For each $x \in Q_{0}$ , step 1.2 gives $L_{x} g = (L_{x} f) * g_{j}$ , so $∥ L_{x} g - g ∥_{1} \leq ∥ (L_{x} f) * g_{j} - g_{j} ∥_{1} + ∥ f * g_{j} - g_{j} ∥_{1} < ε$ . Therefore $Δ_{Q} (g) \leq ε$ , proving (P1) for arbitrary compact $Q$ and positive $ε$ .

5.1F2F3step 4.1construct∎

If $G$ is amenable, let $ν$ be its mean on $L^{\infty} (G)$ from [F3] and define $m (ψ) : = ν ([ψ])$ for $ψ \in U C B (G)$ . By [F2] this is well-defined, positive, complex-linear and unital; the class map intertwines left translations, so $m$ is left invariant. Applying steps 1.1–4.1 gives (P1).

Sources

BHV, Kazhdan's Property (T), Appendix G.3, Theorem G.3.1 (ii) to (iii), printed p. 455, uses strong continuity to make the translate orbit compact and then applies the uniform approximate-invariance net. Thomas, Lecture 20, slides 17–18 (PDF pp. 17–18), gives the same compact-orbit convolution estimate. Both source proofs take a compact set containing $e$ ; the proof above handles arbitrary compact tests by adjoining $e$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

A Reiter net has an invariant-mean cluster point

Statement

Assume the ultrafilter lemma. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , and let $(f_{i})_{i \in I}$ be a Reiter net on $G$ as in Reiter's condition (P1). Define $λ_{f_{i}} (φ) : = \int_{G} f_{i} φ d μ (φ \in L^{\infty} (G)) .$ Then $(λ_{f_{i}})$ has a weak-star cluster point $m$ in $L^{\infty} (G)^{*}$ . Every such cluster point is a left-invariant mean on $L^{\infty} (G)$ , so Reiter's condition (P1) implies that $G$ is amenable (Amenable locally compact group).

Facts & Assumptions

Given: The ultrafilter lemma, a locally compact Hausdorff group $G$ with fixed left Haar measure $μ$ , and a net $(f_{i})$ in the probability densities $P$ satisfying Reiter's compact-uniform translation condition.

[A1]

The ultrafilter lemma is assumed exactly as stated; Banach–Alaoglu and the compactness-to-net-cluster-point implication use this principle (The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter, Banach–Alaoglu, Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging).

[F1]

$L^{\infty} (G)$ is a complex normed space of almost-everywhere classes with the essential-supremum norm; its dual consists of bounded complex-linear functionals, and weak-star convergence is pointwise convergence on $L^{\infty} (G)$ (Complex $L^{\infty}$ space of a locally compact group, The dual space X^* of a normed space and its dual norm, The weak-star topology from finite evaluations).

[F2]

For $f \in P$ , $f \geq 0$ and $\int_{G} f d μ = ∥ f ∥_{1} = 1$ ; for every $φ \in L^{\infty} (G)$ and $η > 0$ , $∣ φ ∣ \leq ∥ φ ∥_{\infty} + η$ almost everywhere. The integral is complex-linear, monotone on nonnegative functions, and satisfies the integral triangle inequality (Reiter's condition (P1), Complex Haar L^p spaces and compactly supported functions, Complex $L^{\infty}$ space of a locally compact group, Integrable real and complex functions, and their integrals, The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F3]

Left translations preserve Haar measure, so the substitution $x = g y$ in a Haar integral is valid; the Reiter net satisfies $∥ L_{h} f_{i} - f_{i} ∥_{1} \to 0$ for every fixed $h \in G$ (Left Haar integral and left Haar measure, Measure-preserving transformations and systems, Integral invariance under measure-preserving maps, Reiter's condition (P1)).

[F4]

The closed dual unit ball of a normed space is weak-star compact under the ultrafilter lemma. In a compact space every net has a cluster point, and each cluster point of a net is the limit of a subnet (Banach–Alaoglu, Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging, A point is a cluster point of a net if and only if some subnet converges to it).

[F5]

A mean on $L^{\infty} (G)$ is a positive complex-linear functional with $m (1_{G}) = 1$ , and it is left-invariant when $m (L_{g} φ) = m (φ)$ for every $g$ and $φ$ (Left-invariant means on $L^{\infty}$ of a locally compact group).

[F6]

Amenability of $G$ means that such a left-invariant mean exists (Amenable locally compact group).

Proof

technique · direct

1.1F1F2

For each $i$ , define $λ_{f_{i}} (φ) = \int_{G} f_{i} φ d μ$ on $L^{\infty} (G)$ . This is independent of representatives because changing either factor on a null set changes the product only almost everywhere. For each $η > 0$ , [F2] and positivity of $f_{i}$ give $∣ f_{i} φ ∣ \leq f_{i} ∣ φ ∣ \leq f_{i} (∥ φ ∥_{\infty} + η)$ almost everywhere, so $f_{i} φ$ is integrable and $∣ λ_{f_{i}} (φ) ∣ \leq \int_{G} f_{i} ∣ φ ∣ d μ \leq (∥ φ ∥_{\infty} + η) ∥ f_{i} ∥_{1} = ∥ φ ∥_{\infty} + η$ . Letting $η ↓ 0$ proves boundedness with norm at most one. Linearity of the integral makes $λ_{f_{i}}$ complex-linear, so it lies in the closed dual unit ball of $L^{\infty} (G)^{*}$ .

1.2A1F1F2F4F5

By [A1] and [F4], the dual unit ball is weak-star compact and the net $(λ_{f_{i}})$ has a weak-star cluster point. Fix any cluster point $m$ ; [F4] supplies a subnet $(λ_{f_{i_{j}}})$ converging weak-star to $m$ . For each $φ \geq 0$ , every $λ_{f_{i_{j}}} (φ) = \int f_{i_{j}} φ d μ$ is nonnegative, so continuity of evaluation in the weak-star topology gives $m (φ) \geq 0$ . Also $λ_{f_{i_{j}}} (1_{G}) = \int f_{i_{j}} d μ = 1$ for every $j$ , hence $m (1_{G}) = 1$ . Since $m \in L^{\infty} (G)^{*}$ already, it is a mean by [F5].

2.1F1F3F5step 1.2

Fix $g \in G$ and $φ \in L^{\infty} (G)$ . Left invariance of Haar measure, with $x = g y$ , gives $λ_{f_{i}} (L_{g} φ) = \int_{G} f_{i} (x) φ (g^{- 1} x) d μ (x) = \int_{G} f_{i} (g y) φ (y) d μ (y) = λ_{L_{g^{- 1}} f_{i}} (φ)$ . Therefore $∣ λ_{f_{i}} (L_{g} φ) - λ_{f_{i}} (φ) ∣ \leq ∥ L_{g^{- 1}} f_{i} - f_{i} ∥_{1} ∥ φ ∥_{\infty} \to 0$ by [F2, F3] and the Reiter condition on the compact singleton ${g^{- 1}}$ . Along the subnet from step 1.2, weak-star convergence makes the left difference converge to $∣ m (L_{g} φ) - m (φ) ∣$ , which must consequently be zero. As $g$ and $φ$ were arbitrary, $m$ is left-invariant by [F5].

3.1F2F6step 1.2step 2.1∎

Steps 1.2 and 2.1 prove that the net has a cluster point and every cluster point is a left-invariant mean. If $G$ satisfies Reiter's condition, its equivalent net formulation [F2] supplies such a Reiter net; the cluster-point mean then witnesses amenability by [F6].

Sources

BHV, Kazhdan's Property (T), Appendix G, Theorem G.3.1, implication (iv) to (v), states that a weak-star limit point of Reiter densities is invariant. Thomas, Lecture 19, slides 5–7, records the $L^{1}$ -to-dual pairing by integration. Daws–Runde, Introduction, printed p. 1 after equation (1), explicitly states that each weak-star accumulation point of an asymptotically invariant $L^{1}$ -probability net is a left-invariant mean. The present proof supplies the complex-functional well-definedness, the exact Haar substitution, and the ultrafilter-lemma assumption at the dual-ball compactness step.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Amenability is equivalent to Reiter's condition (P1)

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group. Then $G$ is amenable (Amenable locally compact group) if and only if $G$ satisfies Reiter's condition (P1) (Reiter's condition (P1)): for every compact $Q \subseteq G$ and every $ε > 0$ there is $f \in L^{1} (G)$ with $f \geq 0$ , $∥ f ∥_{1} = 1$ and $Δ_{Q} (f) \leq ε$ , with $Δ_{Q}$ as defined in Reiter's condition (P1) (including $Δ_{\emptyset} (f) = 0$ ). The equivalence is proved through invariant means on $L^{\infty} (G)$ , so it is stated for a fixed left Haar measure but does not depend on its normalization.

Facts & Assumptions

Given: AC, a locally compact Hausdorff group $G$ , and a fixed left Haar measure $μ$ .

[A1]

AC is assumed in the choice-function form (The Axiom of Choice).

[F1]

The class map embeds actual UCB functions isometrically in $L^{\infty} (G)$ and intertwines the left translations (Left-uniformly continuous bounded functions (UCB)).

[F2]

A left-invariant mean on UCB(G) yields Reiter's condition (P1) under AC (An invariant mean produces a Reiter net).

[F3]

Reiter's condition (P1) is equivalent to the existence of a net in $P$ with compact-uniform translation defects (Reiter's condition (P1)).

[F4]

Amenability is existence of a left-invariant mean on complex $L^{\infty} (G)$ and is invariant under positive rescaling of Haar measure (Amenable locally compact group, Left-invariant means on $L^{\infty}$ of a locally compact group).

[F5]

AC implies the ultrafilter lemma (The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter).

[F6]

Under the ultrafilter lemma, every Reiter net has a weak-star cluster point which is a left-invariant mean on $L^{\infty} (G)$ (A Reiter net has an invariant-mean cluster point).

[F7]

Multiplying a left Haar measure by a positive scalar preserves left invariance and the Haar measure convention (Left Haar integral and left Haar measure).

Proof

technique · direct

1.1F1F2F4construct

Suppose $G$ is amenable. Let $ν$ be a left-invariant mean on $L^{\infty} (G)$ and define $m (ψ) : = ν ([ψ])$ for $ψ \in U C B (G)$ . By [F1], the class map is well-defined and injective; positivity, complex linearity, normalization, and left invariance pass from $ν$ to $m$ . Hence [F2] gives Reiter's condition (P1).

1.2A1F3F4F5F6

Suppose $G$ satisfies (P1). By [F3], choose a Reiter net $(f_{i}) \subseteq P$ . By [A1] and [F5], the ultrafilter lemma holds. The cluster-point result [F6] gives a left-invariant mean on $L^{\infty} (G)$ , so $G$ is amenable by [F4].

1.3F3F4F7algebra

Let $μ^{'} = c μ$ for $c > 0$ . The measures have the same null sets, and $f \mapsto f^{'} : = c^{- 1} f$ maps $P_{μ}$ bijectively to $P_{μ^{'}}$ . For every compact $Q$ , $L_{x} f^{'} = c^{- 1} L_{x} f$ on the common almost-everywhere classes, so $Δ_{Q, μ^{'}} (f^{'}) = Δ_{Q, μ} (f)$ . Thus (P1) is independent of this rescaling. Amenability is likewise normalization-independent by [F4], with $μ^{'}$ still a left Haar measure by [F7].

2.1step 1.1step 1.2step 1.3∎

Steps 1.1 and 1.2 prove the two implications for the fixed left Haar measure, and step 1.3 proves normalization independence. Therefore the stated equivalence holds.

Sources

BHV, Kazhdan's Property (T), Appendix G.3, Theorem G.3.1, gives the amenability, Reiter (P1), and invariant-mean equivalence, printed pp. 452–456. The proof here uses the assigned local UCB-to-Reiter and Reiter-cluster-point lemmas rather than its separate weak-star-density assertion for all $L^{1}$ probability densities. Thomas, Lecture 20, slides 11–17 (PDF pp. 11–17), provides the amenability/invariant-mean-to-Reiter direction.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Følner nets give Reiter nets

Statement

Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , and let $F \subseteq G$ be Borel with $0 < μ (F) < \infty$ . The class $f_{F} : = μ (F)^{- 1} 1_{F} \in L^{1} (G)$ belongs to $P$ from Reiter's condition (P1), and for every $x \in G$ , $∥ L_{x} f_{F} - f_{F} ∥_{1} = \frac{μ (x F △ F)}{μ (F)} .$ Consequently, every left Følner net $(F_{i})$ gives a Reiter net $(f_{F_{i}})$ . In particular, the left Følner condition implies Reiter's condition (P1).

Facts & Assumptions

Given: A locally compact Hausdorff group $G$ with left Haar measure $μ$ and a Borel set $F$ with $0 < μ (F) < \infty$ .

[A1]

Left translation carries Borel sets to Borel sets and preserves $μ$ ; thus $μ (x F) = μ (F)$ and $μ (x F △ F) \leq 2 μ (F)$ (Left Haar integral and left Haar measure, A continuous map has Borel preimages of Borel sets).

[F1]

$L^{1} (G)$ consists of complex measurable almost-everywhere classes with $∥ f ∥_{1} = \int_{G} ∣ f ∣ d μ$ (Complex Haar L^p spaces and compactly supported functions).

[F2]

An indicator of a Borel set is a nonnegative simple measurable function; its nonnegative Lebesgue integral is its simple integral, namely the measure of that set (A measurable function between measurable spaces, Nonnegative simple measurable functions, The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions).

Proof

technique · direct

1.1F1F2givenconstructalgebra

The indicator $1_{F}$ is Borel measurable and simple. By [F2], $\int_{G} 1_{F} d μ = μ (F)$ , so the nonnegative Borel function $μ (F)^{- 1} 1_{F}$ has finite integral and defines an $L^{1} (G)$ class. It is nonnegative and $∥ f_{F} ∥_{1} = μ (F)^{- 1} \int_{G} 1_{F} d μ = 1$ ; hence $f_{F} \in P$ .

1.2A1F1F2algebra

For every $x, y \in G$ , $(L_{x} 1_{F}) (y) = 1_{F} (x^{- 1} y) = 1_{x F} (y)$ . Thus $L_{x} f_{F} - f_{F} = μ (F)^{- 1} (1_{x F} - 1_{F})$ , whose modulus is $μ (F)^{- 1} 1_{x F △ F}$ . The symmetric difference is Borel and has finite measure by [A1]. Applying [F2] to its indicator gives $∥ L_{x} f_{F} - f_{F} ∥_{1} = μ (F)^{- 1} \int_{G} 1_{x F △ F} d μ = \frac{μ (x F △ F)}{μ (F)}$ .

2.1step 1.1step 1.2givenconstruct∎

If $(F_{i})$ is a left Følner net, set $f_{i} : = f_{F_{i}}$ . Steps 1.1 and 1.2 show that $f_{i} \in P$ and, for every compact $Q$ , $Δ_{Q} (f_{i}) = Δ_{Q} (F_{i})$ , since the pointwise discrepancies agree for each $x \in Q$ . The defining eventual estimates therefore make $(f_{i})$ a Reiter net. If only the single-set left Følner condition is given, for each compact $Q$ and $ε > 0$ choose a Følner witness $F$ ; step 1.1 gives $f_{F} \in P$ and step 1.2 gives the same estimate, so Reiter's condition (P1) holds. This uses one witness at a time and no global choice function.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The layer-cake identity for integrable functions

Statement

Assume the Axiom of Countable Choice. Let $(X, A, μ)$ be any measure space, and let $f, g \in L^{1} (X)$ be nonnegative integrable functions with fixed pointwise nonnegative measurable representatives. For $t > 0$ put $E_{t} : = {x \in X : f (x) \geq t}$ and $E_{t}^{'} : = {x \in X : g (x) \geq t}$ , and let $d t$ denote Lebesgue measure on the level parameter. Then $∥ f - g ∥_{1} = \int_{0}^{\infty} μ (E_{t} △ E_{t}^{'}) d t,$ and in particular $∥ f ∥_{1} = \int_{0}^{\infty} μ (E_{t}) d t .$ No $σ$ -finiteness of $μ$ is required: both integrands are supported on $S : = {x : f (x) + g (x) > 0}$ , which is $σ$ -finite.

Facts & Assumptions

Given: The Axiom of Countable Choice, a measure space $(X, A, μ)$ , and nonnegative $L^{1}$ classes with the fixed representatives in the statement.

[A1]

The Axiom of Countable Choice is the assumption used to construct the library's Lebesgue measure on $R$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$L^{1} (μ)$ is a real vector space, its quotient $L^{1} (μ)$ uses almost-everywhere classes, and its norm is the integral of the absolute value (The function space $L^{p} (μ)$ for $0 < p < \infty$ , The space $L^{p} (μ)$ as the quotient by null functions, $L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$ ).

[F2]

For a nonnegative measurable $h$ and $a > 0$ , $μ ({h \geq a}) \leq a^{- 1} \int h d μ$ (Chebyshev-Markov inequality for the integral).

[F3]

Under [A1], Lebesgue measure $λ$ on $R$ is a measure, is $σ$ -finite, and assigns length $b - a$ to each interval $(a, b]$ (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume, Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure, A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Finite, sigma-finite, and semifinite measures).

[F4]

The rationals are countable and dense in $R$ , and for every $ε > 0$ some $n \in N$ has $1 / n < ε$ ( $Q$ is countably infinite, Every subset of an at most countable set is at most countable, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable, For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ , The natural numbers $N$ (von Neumann)).

[F5]

Product-measurable rectangles, countable unions and intersections are measurable; product measure is defined for $σ$ -finite measure spaces and Tonelli's theorem interchanges the integrals of a nonnegative product-measurable function on such a product (Sigma-algebras, A measurable function between measurable spaces, The Borel sigma-algebra of a topological space, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The product measure of two sigma-finite measure spaces, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

Proof

technique · direct

1.1F1F2F4

Put $h : = f + g$ and $S : = {x : h (x) > 0}$ . By [F1], $h \in L^{1} (μ)$ and $\int h d μ < \infty$ . For each $n \geq 1$ , $S_{n} : = {x : h (x) \geq 1 / n}$ is measurable and [F2] gives $μ (S_{n}) \leq n \int h d μ < \infty$ . The sets $S_{n}$ cover $S$ : if $h (x) > 0$ , [F4] gives $n$ with $1 / n < h (x)$ . Thus $S$ is $σ$ -finite with its restricted measure, and $f = g = 0$ on $X ∖ S$ .

1.2F3F4F5algebra

For $f$ , define $A_{f} : = {(x, t) \in S \times R : 0 < t \leq f (x)}$ . Countability of $Q_{> 0}$ and [F5] make $A_{f} = ⋂_{m \geq 1} ⋃_{q \in Q_{> 0}} ((S \cap {x : f (x) > q}) \times (0, q + 1 / m])$ product-measurable. Indeed, if $0 < t \leq f (x)$ , density of $Q$ supplies for each fixed $m$ a rational $q$ with $\max (0, t - 1 / m) < q < f (x)$ , so $t \leq q + 1 / m$ . Conversely, membership in every union gives $f (x) > t - 1 / m$ for each $m$ ; if $f (x) < t$ , [F4] gives $m$ with $1 / m < t - f (x)$ , a contradiction. Define $A_{g}$ in the same way and put $D : = A_{f} △ A_{g}$ . For each $t > 0$ , its section is $E_{t} △ E_{t}^{'}$ , while for each $x \in S$ its section is $(0, f (x)] △ (0, g (x)]$ , whose Lebesgue measure is $∣ f (x) - g (x) ∣$ by [F3].

2.1A1F1F3F5step 1.1step 1.2

The restricted measure $μ ∣_{S}$ is $σ$ -finite by step 1.1, and $λ$ is $σ$ -finite by [F3], so [F5] applies to $1_{D}$ on their product. Since $D \subseteq S \times (0, \infty)$ and $f = g = 0$ off $S$ , Tonelli gives $∥ f - g ∥_{1} = \int_{S} ∣ f - g ∣ d μ = \int_{S} \int_{R} 1_{D} (x, t) d λ (t) d μ (x) = \int_{R} \int_{S} 1_{D} (x, t) d μ (x) d λ (t) = \int_{0}^{\infty} μ (E_{t} △ E_{t}^{'}) d t$ .

3.1step 2.1∎

Taking $g = 0$ in step 2.1 gives $∥ f ∥_{1} = \int_{0}^{\infty} μ (E_{t}) d t$ . The only choice assumption used is [A1]; the reduction from $X$ to $S$ and the countable product-measurability description are explicit.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Reiter functions can be cut down to Følner sets

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , let $Q \subseteq G$ be compact with $μ (Q) > 0$ , let $ε > 0$ , and let $f \in L^{1} (G)$ satisfy $f \geq 0$ , $∥ f ∥_{1} = 1$ and $\sup_{x \in Q^{2}} ∥ L_{x} f - f ∥_{1} \leq \frac{ε μ (Q)}{2 μ (Q^{2})},$ where $Q^{2} : = {x y : x, y \in Q}$ . Then there is a Borel set $U \subseteq G$ with $0 < μ (U) < \infty$ and $\sup_{x \in Q} \frac{μ (x U △ U)}{μ (U)} \leq 2 ε .$ Consequently Reiter's condition (P1) implies the left Følner condition: for every compact $Q$ and every $ε > 0$ a Borel set $U$ with $0 < μ (U) < \infty$ and $\sup_{x \in Q} \frac{μ (x U △ U)}{μ (U)} \leq 2 ε$ exists. The supremum over an empty compact test set in the consequent is taken to be $0$ .

Facts & Assumptions

Given: AC; an LCH group $G$ with fixed left Haar measure $μ$ ; a compact $Q$ with $μ (Q) > 0$ ; $ε > 0$ ; and a nonnegative norm-one $f \in L^{1} (G)$ satisfying the Statement's $Q^{2}$ estimate.

[F1]

Left translations $L_{a}$ are linear isometries on $L^{1} (G)$ , satisfy $L_{a} L_{b} = L_{a b}$ and have norm-continuous vector orbits under AC (Strong continuity of left and modular right translations on L1 and L2).

[F2]

Under AC, complex $L^{1} (G)$ is complete. Integrals are linear, obey the integral triangle inequality and are monotone on nonnegative functions; a nonnegative function has zero integral exactly when it is zero a.e. (Completeness of the complex Haar L1 and L2 spaces and density of Cc, The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, Complex Haar L^p spaces and compactly supported functions).

[F3]

Left Haar measure is left invariant, finite on compact sets and positive on nonempty opens; every identity has a compact neighborhood. Finite products and continuous images preserve compactness (Left Haar integral and left Haar measure, Haar measure is positive on nonempty open sets and finite on compact sets, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Topological group: multiplication and inversion are continuous, A product of finitely many compact spaces is compact in the product topology, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F4]

Under Countable Choice, layer cake gives $\int_{0}^{\infty} μ ({u \geq t}) d t = ∥ u ∥_{1}$ and the analogous symmetric-difference identity for two nonnegative integrable functions, without global sigma-finiteness. The same proof with intervals $(0, u)$ instead of $(0, u]$ gives the strict-superlevel version; their endpoints have zero Lebesgue length (The layer-cake identity for integrable functions).

[F5]

Chebyshev bounds $μ ({u > t}) \leq ∥ u ∥_{1} / t$ for $u \geq 0$ , $t > 0$ . Tonelli interchanges nonnegative product-measurable integrals on sigma-finite spaces, and pointwise limits of measurable functions are measurable (Chebyshev-Markov inequality for the integral, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable).

[F6]

Compact subsets of a Hausdorff space are closed, hence Borel, and a finite open cover can be disjointified into Borel cells by finite differences. Compactness gives finite subcovers (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, The Borel sigma-algebra of a topological space).

[F7]

Reiter (P1) supplies a probability density for every compact test and positive tolerance; the left Følner condition uses finite-positive Borel sets and compact-uniform boundary defects (Reiter's condition (P1), Left Følner nets for locally compact groups).

[A1]

AC implies Countable Choice by the declared implication, supplying the hypothesis in [F4]; AC also chooses the finite-partition data for each positive integer below (The Axiom of Choice, AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · parity-average the supplied density, then combine scalar left-Haar averaging with measurable coarea to select a single uniform Følner level set

1.1F1F2F3F6A1givenconstructalgebra

Put $K = Q^{2}$ , $q = μ (Q)$ and $k = μ (K)$ . By [F3, F6], $K$ is compact Borel, $k < \infty$ , and for any $c \in Q$ , $c Q \subseteq K$ gives $k \geq q > 0$ . Let $δ = \sup_{b \in K} ∥ L_{b} f - f ∥_{1}$ , which is finite by the bound $2 ∥ f ∥_{1} = 2$ , and satisfies $δ \leq ε q / (2 k)$ . Construct the compact-orbit probability average directly in $L^{1}$ : for each $n$ , cover the compact orbit by norm balls of radius $1 / (2 n)$ , pull them back to a finite open cover of $Q$ , and disjointify by [F6]. Choose a sample from each nonempty cell; this gives a finite Borel partition $Q = ⨆_{j} E_{n, j}$ with samples $y_{n, j} \in E_{n, j}$ such that $∥ L_{y} f - L_{y_{n, j}} f ∥_{1} < 1 / n$ on each nonempty cell. Set $S_{n} = \sum_{j} μ (E_{n, j}) L_{y_{n, j}} f / q$ . Intersecting the partitions for $n, m$ and comparing their samples through a point of each nonempty intersection gives $∥ S_{n} - S_{m} ∥_{1} \leq 1 / n + 1 / m$ . Completeness in [F2] gives a limit $g$ . Every $S_{n}$ is a nonnegative probability density; convergence makes the imaginary part and negative real part of $g$ have zero $L^{1}$ norm, hence $g \geq 0$ , and norm continuity gives $∥ g ∥_{1} = 1$ . This compact finite-partition mean requires no pointwise formula for extended convolution.

2.1F1F2step 1.1constructalgebra

For each $b \in Q$ , $b y_{n, j} \in K$ , so $∥ L_{b} S_{n} - f ∥_{1} \leq δ$ and passage to the norm limit gives $∥ L_{b} g - f ∥_{1} \leq δ$ . For $a \in Q$ , fix any $c \in Q$ ; [F1] gives $∥ L_{a} f - g ∥_{1} = ∥ L_{c} L_{a} f - L_{c} g ∥_{1} \leq ∥ L_{c a} f - f ∥_{1} + ∥ f - L_{c} g ∥_{1} \leq 2 δ$ . Thus $h = (f + g) / 2$ is a probability density with $∥ L_{a} h - h ∥_{1} \leq 3 δ / 2$ for $a \in Q$ . For $x = a b \in K$ , $∥ L_{x} g - g ∥_{1} \leq ∥ L_{a} (L_{b} g - f) ∥_{1} + ∥ L_{a} f - g ∥_{1} \leq 3 δ$ ; together with $∥ L_{x} f - f ∥_{1} \leq δ$ , this gives $∥ L_{x} h - h ∥_{1} \leq 2 δ$ on $K$ . No factors were commuted.

2.2F1F2F3step 1.1algebra

For any finite-positive Borel $U$ , write $d_{a} (U) = μ (a U △ U) = ∥ L_{a} 1_{U} - 1_{U} ∥_{1}$ and $C_{P} (U) = \int_{P} d_{a} (U) d μ (a)$ for $P = Q, K$ . For every $x, a \in Q$ , insert $L_{x} L_{a} 1_{U}$ and use [F1] to obtain $d_{x} (U) \leq d_{a} (U) + d_{x a} (U)$ . Integrating over $a \in Q$ , left invariance and $x Q \subseteq K$ give $q d_{x} (U) \leq C_{Q} (U) + \int_{x Q} d_{b} (U) d μ (b) \leq C_{Q} (U) + C_{K} (U)$ . Therefore $\sup_{x \in Q} d_{x} (U) / μ (U) \leq (C_{Q} (U) + C_{K} (U)) / (q μ (U))$ . This scalar averaging inequality needs no identity in $Q$ , inverse-word cover, overlap inference or right-Haar factor.

3.1F1F3F5F6step 2.1constructalgebra

Choose a nonnegative finite-valued Borel representative of $h$ and put $U_{t} = {h > t}$ for $t > 0$ . By [F5], $H (t) = μ (U_{t}) \leq 1 / t < \infty$ ; it is nonincreasing and hence Borel measurable. If $s ↓ t$ , the sets $U_{s}$ increase to $U_{t}$ , so countable additivity gives $∥ 1_{U_{s}} - 1_{U_{t}} ∥_{1} \to 0$ . For each fixed $t$ , $D (a, t) = d_{a} (U_{t})$ is continuous in $a$ by [F1]. To prove product measurability, set $t_{n} (t) = 2^{- n} (⌊ 2^{n} t ⌋ + 1) > t$ , which decreases to $t$ . For each positive integer $j$ , $t_{n} (t) = j 2^{- n}$ on $[(j - 1) 2^{- n}, j 2^{- n}) \cap (0, \infty)$ ; thus a strict superlevel set of $D (a, t_{n} (t))$ is the countable union of the products of ${a : D (a, j 2^{- n}) > c}$ with these Borel intervals. The $a$ -sets are open by [F1], so $D (a, t_{n} (t))$ is product-measurable. The estimate $∣ D (a, t_{n} (t)) - D (a, t) ∣ \leq 2 ∥ 1_{U_{t_{n} (t)}} - 1_{U_{t}} ∥_{1}$ tends to zero uniformly in $a$ by [F1], so [F5] gives product measurability of $D$ . This constructs the bridge even when $G$ is not second countable; it does not treat arbitrary Borel functions on products as product-measurable.

4.1F2F3F4F5step 2.1step 3.1choosealgebra

For each $a$ , the strict-superlevel layer-cake identity [F4] gives $\int_{0}^{\infty} D (a, t) d t = ∥ L_{a} h - h ∥_{1}$ , while $\int_{0}^{\infty} H (t) d t = 1$ . Haar measure restricted to $Q$ and $K$ is finite; the level measure is sigma-finite, so product measurability from step 3.1 permits [F5] on each restricted product. For $F (t) = C_{Q} (U_{t}) + C_{K} (U_{t})$ this yields $\int_{0}^{\infty} F (t) d t \leq B : = δ (3 q / 2 + 2 k)$ by step 2.1. When $H (t) = 0$ , left invariance gives $F (t) = 0$ . There is a $t > 0$ with $H (t) > 0$ and $F (t) \leq B H (t)$ : otherwise the nonnegative measurable difference $F - B H$ would be strictly positive on ${H > 0}$ , a set of positive level measure since $\int H = 1$ , contradicting $\int (F - B H) \leq 0$ and [F2]. This also handles $δ = B = 0$ . Choose that $t$ and set $U = U_{t}$ .

5.1step 1.1step 2.2step 4.1algebra

With $r = k / q \geq 1$ , steps 2.2 and 4.1 give $\sup_{x \in Q} μ (x U △ U) / μ (U) \leq B / q = δ (3 / 2 + 2 r) \leq ε (1 + 3 / (4 r)) \leq 7 ε / 4 < 2 ε$ . Also $U$ is Borel and $0 < μ (U) < \infty$ by step 4.1. This proves the full original quantitative claim, including $δ = 0$ , with the stronger derived bound $7 ε / 4$ ; the stronger bound is a local conclusion, not an assertion about the cited source's identity-containing route.

6.1F3F6F7step 5.1construct∎

Finally assume (P1) and fix any compact target $T$ and $ε > 0$ . Choose a compact identity neighborhood $C$ by [F3] and put $Q_{0} = T \cup C$ . It is compact and has positive finite measure by [F3, F6]; so does $Q_{0}^{2}$ . Apply [F7] on $Q_{0}^{2}$ with tolerance $ε μ (Q_{0}) / (2 μ (Q_{0}^{2}))$ , and apply the just-proved quantitative clause to this density and $Q_{0}$ . The resulting $U$ satisfies the promised $2 ε$ bound on $Q_{0}$ , hence on $T$ . Empty $T$ has defect zero under the stated convention. Since the requested tolerance can also be replaced by half of any desired Følner tolerance, this is the full left Følner condition. No compact generation, countability or semifiniteness was used.

Remarks

The source extraction proofs begin with $e \in Q$ . Their overlap inference $μ (x Q^{2} \cap Q^{2}) \geq μ (Q)$ is false for an unqualified $Q$ : on the additive real line, $Q = [1, 2]$ and $x = 3 / 2$ give $Q^{2} = [2, 4]$ and $(x + Q^{2}) \cap Q^{2} = [7 / 2, 4]$ , of half the measure of $Q$ . Also for $Q = [100, 101]$ every $A \subseteq Q^{2} = [200, 202]$ has $A - A \subseteq [- 2, 2]$ , so $Q \subseteq A A^{- 1}$ is impossible. These examples refute that route, not the quantitative conclusion. The local proof above preserves the arbitrary-positive-compact- $Q$ claim by scalar left-Haar averaging and coarea after parity symmetrization, replacing the invalid overlap route without adding $e \in Q$ or changing the repository's Haar/null conventions.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The Følner criterion for locally compact groups

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ . Then $G$ is amenable (Amenable locally compact group) if and only if it satisfies the left Følner condition (Left Følner nets for locally compact groups): for every compact $Q \subseteq G$ and every $ε > 0$ there is a Borel set $F \subseteq G$ with $0 < μ (F) < \infty$ and $Δ_{Q} (F) \leq ε$ . Equivalently, $G$ admits a left Følner net. Borel sets suffice for $F$ .

Facts & Assumptions

Given: AC, a locally compact Hausdorff group $G$ , and a fixed left Haar measure $μ$ .

[A1]

AC is the choice-function principle (The Axiom of Choice). It implies AC $ω$ : for any sequence $(X_{n})$ of nonempty sets, apply AC to the range family ${X_{n} : n \in N}$ and use the resulting selector at each $X_{n}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Amenability is equivalent to Reiter's condition (P1) for locally compact Hausdorff groups under AC; (P1) means compact-uniform approximate invariance of nonnegative norm-one $L^{1}$ functions (Amenability is equivalent to Reiter's condition (P1), Reiter's condition (P1)).

[F2]

For nonnegative $f \in L^{1} (G)$ , its superlevel sets $E_{t} = {y : f (y) \geq t}$ satisfy $\int_{0}^{\infty} μ (E_{t}) d t = ∥ f ∥_{1}$ and $\int_{0}^{\infty} μ (E_{t} △ E_{t}^{'}) d t = ∥ f - g ∥_{1}$ for any nonnegative $g \in L^{1} (G)$ with corresponding superlevel sets $E_{t}^{'}$ ; no global $σ$ -finiteness of Haar measure is required (The layer-cake identity for integrable functions).

[F3]

If $f \geq 0$ and $∥ f ∥_{1} = 1$ , then $μ ({f \geq t}) \leq 1 / t$ for every $t > 0$ (Chebyshev-Markov inequality for the integral).

[F4]

Left Haar measure is left invariant, finite on compact sets, and positive on nonempty open sets (Left Haar integral and left Haar measure, Haar measure is positive on nonempty open sets and finite on compact sets).

[F5]

Under AC, $x \mapsto L_{x} u$ is norm-continuous in $L^{1} (G)$ for each $u \in L^{1} (G)$ , and every $L_{x}$ is an isometry (Strong continuity of left and modular right translations on L1 and L2).

[F6]

Every identity in a locally compact Hausdorff group has a compact neighbourhood; finite products of compact spaces are compact, and continuous images of compact spaces are compact (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, A product of finitely many compact spaces is compact in the product topology, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Topological group: multiplication and inversion are continuous).

[F7]

Tonelli interchanges nonnegative integrals on a product of $σ$ -finite measure spaces, and pointwise limits of measurable real functions are measurable (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable).

[F8]

For a Borel set $F$ with $0 < μ (F) < \infty$ , the normalized indicator $μ (F)^{- 1} 1_{F}$ is a Reiter probability density with translation defect exactly $μ (x F △ F) / μ (F)$ (Følner nets give Reiter nets).

[F9]

A net of positive finite-measure Borel sets satisfies the left Følner condition exactly when the single-set condition in the Statement holds (Left Følner nets for locally compact groups).

[F10]

A compact subset of a Hausdorff space is closed, hence Borel (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, The Borel sigma-algebra of a topological space).

[F11]

$L^{1} (G)$ is formed from measurable functions for the fixed Borel Haar measure; every class therefore has a Borel representative, and replacing a representative by its positive part preserves its class when the class is nonnegative (Complex Haar L^p spaces and compactly supported functions, Left Haar integral and left Haar measure, The Borel sigma-algebra of a topological space).

[F12]

Amenability of $G$ means existence of a left-invariant mean on $L^{\infty} (G)$ (Amenable locally compact group).

Proof

Given: AC and a locally compact Hausdorff group $G$ with fixed left Haar measure $μ$ .

Proof technique: direct.

1.1F1F12given

Suppose $G$ is amenable in the sense of [F12]. By [F1], $G$ satisfies Reiter's condition (P1).

1.2F1F4F6F10givenconstruct

Assume (P1), fix a compact target $Q \subseteq G$ and $ε > 0$ , and put $η : = ε / 2$ . Choose a compact neighbourhood $C$ of the identity and an open identity neighbourhood $O \subseteq C$ . Set $P : = Q \cup C$ and $K : = P^{2}$ . Then $P$ is compact, contains the identity, and $0 < μ (P) < \infty$ because $O \subseteq P$ and [F4]; $K$ is compact and Borel by [F6] and [F10], and $0 < μ (K) < \infty$ because $P \subseteq K$ . Choose $f \in L^{1} (G)$ with $f \geq 0$ , $∥ f ∥_{1} = 1$ and $\sup_{x \in K} ∥ L_{x} f - f ∥_{1} \leq η μ (P) / (4 μ (K))$ , using (P1) with this positive tolerance.

1.3F1F8F9F12

Conversely, suppose $G$ satisfies the left Følner condition. By [F8], normalized indicators of its Følner witnesses give (P1) (equivalently use a Følner net by [F9]); [F1] then implies that $G$ is amenable in the sense of [F12].

2.1A1F2F3F4F5F7F8F11step 1.2algebra

By [F11] choose a nonnegative Borel representative of $f$ and set $E_{t} : = {y : f (y) \geq t}$ for $t > 0$ . By [F2] and [F3], $\int_{0}^{\infty} μ (E_{t}) d t = 1$ and $μ (E_{t}) < \infty$ for every $t > 0$ . For each $t$ with $μ (E_{t}) > 0$ , [F8] applied to $E_{t}$ gives $μ (x E_{t} △ E_{t}) = ∥ L_{x} 1_{E_{t}} - 1_{E_{t}} ∥_{1}$ ; when $μ (E_{t}) = 0$ , both sides vanish by [F4]. Thus the boundary function is continuous in $x$ by [F5]. Put $D (x, t) : = ∥ L_{x} 1_{E_{t}} - 1_{E_{t}} ∥_{1}$ . On each level interval $[1 / n, n]$ , for all sufficiently large $m$ put $t_{m} (t) : = 2^{- m} ⌊ 2^{m} t ⌋ > 0$ . Then $t_{m} (t) ↑ t$ , so the finite-measure sets $E_{t_{m} (t)}$ decrease to $E_{t}$ and $∥ 1_{E_{t_{m} (t)}} - 1_{E_{t}} ∥_{1} \to 0$ by countable additivity. For fixed $m$ , $D (x, t_{m} (t))$ is product-measurable: it has countably many Borel level-parameter cells, and on each cell is a continuous function of $x$ by [F5]. Isometry gives $∣ D (x, t_{m} (t)) - D (x, t) ∣ \leq 2 ∥ 1_{E_{t_{m} (t)}} - 1_{E_{t}} ∥_{1}$ , so taking the pointwise limit proves product measurability on $K \times [1 / n, n]$ . These intervals cover $(0, \infty)$ , proving the needed product measurability without second countability. Since Haar measure restricted to $K$ is finite and the level parameter has $σ$ -finite Lebesgue measure, [F7] and [F2] give $\int_{0}^{\infty} μ (E_{t}) r (t) d t = \int_{K} ∥ L_{x} f - f ∥_{1} d μ (x) \leq η μ (P) / 4$ , where $r (t) : = \int_{K} μ (x E_{t} △ E_{t}) / μ (E_{t}) d μ (x)$ when $μ (E_{t}) > 0$ , and $r (t) : = 0$ otherwise. Since $\int_{0}^{\infty} μ (E_{t}) d t = 1$ , some $t > 0$ has $0 < μ (E_{t}) < \infty$ and $r (t) < η μ (P) / 2$ .

3.1F3F4F5F10step 2.1algebra

Fix such a $t$ and put $A : = {x \in K : μ (x E_{t} △ E_{t}) / μ (E_{t}) \leq η}$ . The boundary function is continuous, so $A$ is Borel; Markov's inequality gives $μ (K ∖ A) \leq r (t) / η < μ (P) / 2$ . For any $x \in P$ , $x P \subseteq x K \cap K$ because $P \subseteq K$ and $P^{2} = K$ , so $μ (x K \cap K) \geq μ (x P) = μ (P)$ . Also $x K \cap K \subseteq (x A \cap A) \cup x (K ∖ A) \cup (K ∖ A)$ , hence $μ (x A \cap A) > 0$ . Therefore there exist $a_{1}, a_{2} \in A$ with $x = a_{1} a_{2}^{- 1}$ . By left invariance and the triangle inequality for symmetric difference, $μ (x E_{t} △ E_{t}) \leq μ (a_{2}^{- 1} E_{t} △ E_{t}) + μ (a_{1} E_{t} △ E_{t}) = μ (a_{2} E_{t} △ E_{t}) + μ (a_{1} E_{t} △ E_{t}) \leq 2 η μ (E_{t}) = ε μ (E_{t})$ . Thus $F : = E_{t}$ is Borel with positive finite measure and satisfies the required estimate for every $x \in P$ , hence $Δ_{Q} (F) \leq ε$ .

4.1A1F1F2F5F9F12step 1.1step 1.2step 1.3step 2.1step 3.1∎

Steps 1.1, 1.2, 2.1, and 3.1 prove amenability implies the Følner condition; step 1.3 proves the reverse implication. Step 3.1 produces Borel witnesses even when the condition is initially phrased with measurable sets, and [F9] gives the equivalent net formulation. The only Choice assumption is the stated AC, used through [F1], [F5], and AC $ω$ for [F2].

Sources

BHV, Kazhdan's Property (T), Appendix G.5, Theorem G.5.1 and its proof, states the Følner criterion and gives the complete Reiter-to-Følner level-set extraction for a compact test set containing the identity. The proof here enlarges every target compact set to a compact identity neighbourhood, ensuring the positive finite Haar measure required in the averaging estimates, and justifies the compact-parameter Tonelli step under arbitrary LCH generality. Thomas, Lecture 19, slides 14–18, gives the same extraction route.

CorollaryStatement: Literature-sourcedProof: AI-adaptedaudited 2026-10-08Open item page →

Folner sequences for second countable compactly generated groups

Statement

Assume AC. Let $G$ be an amenable second countable compactly generated locally compact Hausdorff group (Amenable locally compact group, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) with fixed left Haar measure $μ$ (Left Haar integral and left Haar measure), and let $S \subseteq G$ be a compact generating set (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Topological group: multiplication and inversion are continuous) with $S = S^{- 1}$ containing $e$ and a nonempty open set. Then there is a sequence $(F_{n})_{n \geq 0}$ of Borel sets with $0 < μ (F_{n}) < \infty$ such that for every compact $Q \subseteq G$ $\lim_{n \to \infty} Δ_{Q} (F_{n}) = 0, Δ_{Q} (F) : = \sup ({0} \cup {μ (x F △ F) / μ (F) : x \in Q}) .$ Thus $Δ_{\emptyset} (F) = 0$ . Such a sequence is a Følner sequence for $G$ . Conversely, a locally compact Hausdorff group with such a sequence satisfies the left Følner condition (Left Følner nets for locally compact groups) and hence is amenable, so for such a group amenability is equivalent to the existence of a Følner sequence. Only compact generation, not second countability, is used by the construction.

Facts & Assumptions

Given: AC; a locally compact Hausdorff group $G$ with fixed left Haar measure $μ$ ; the amenability of $G$ ; a compact generating set $S \subseteq G$ with $S = S^{- 1}$ containing $e$ and a nonempty open set $U \subseteq S$ , with $G = ⋃_{n \geq 1} S^{n}$ .

[A1]

AC implies AC $_{ω}$ : for a sequence $(X_{n})_{n \in N}$ of nonempty sets one applies AC to the family ${X_{n} : n \in N}$ and evaluates the resulting selector at each $X_{n}$ (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Under AC, $G$ is amenable, meaning that there is a left-invariant mean $m$ on $L^{\infty} (G)$ (Amenable locally compact group), if and only if $G$ satisfies the left Følner condition: for every compact $Q \subseteq G$ and every $ε > 0$ there is a Borel set $F \subseteq G$ with $0 < μ (F) < \infty$ and $Δ_{Q} (F) \leq ε$ (The Følner criterion for locally compact groups).

[F2]

For a Borel set $F \subseteq G$ with $0 < μ (F) < \infty$ and compact $Q \subseteq G$ one has $Δ_{Q} (F) = \sup ({0} \cup {μ (x F △ F) / μ (F) : x \in Q})$ , so in particular $Δ_{\emptyset} (F) = 0$ (Left Følner nets for locally compact groups).

[F3]

Multiplication $(x, y) \mapsto x y$ and inversion $x \mapsto x^{- 1}$ are continuous on $G$ ; finite products of compact spaces are compact; and continuous images of compact sets are compact (Topological group: multiplication and inversion are continuous, A product of finitely many compact spaces is compact in the product topology, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F4]

Every ambient open cover of a compact subspace has a finite subcover (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it).

Proof

Given: AC; a locally compact Hausdorff group $G$ with fixed left Haar measure $μ$ ; the amenability of $G$ ; a compact generating set $S = S^{- 1} ∋ e$ containing the nonempty open set $U \subseteq S$ , with $G = ⋃_{n \geq 1} S^{n}$ .

Proof technique: direct.

1.1F3given

For $n \geq 1$ put $K_{n} : = S^{n} = {s_{1} \dots s_{n} : s_{1}, \dots, s_{n} \in S}$ . The map $(s_{1}, \dots, s_{n}) \mapsto s_{1} \dots s_{n}$ is continuous from the finite product $S \times \dots \times S$ , which is compact, onto $K_{n}$ , so $K_{n}$ is compact; and $K_{n} \subseteq K_{n + 1}$ because $e \in S$ .

1.2F1F2algebra

For the reverse implication suppose that $(F_{n})_{n \geq 0}$ is a sequence of Borel sets with $0 < μ (F_{n}) < \infty$ for which $Δ_{Q} (F_{n})$ tends to $0$ for every compact $Q \subseteq G$ . Given a compact $Q$ and $ε > 0$ , the convergence to $0$ gives $n_{1}$ with $Δ_{Q} (F_{n}) \leq ε$ for every $n \geq n_{1}$ ; then $F : = F_{n_{1}}$ is Borel with $0 < μ (F) < \infty$ and $Δ_{Q} (F) \leq ε$ by [F2]. Thus $G$ satisfies the left Følner condition, and [F1] makes $G$ amenable.

2.1F3F4step 1.1givenconstruct

Put $W : = U \cdot U^{- 1}$ . Then $W$ is open, being a union of translates of the open set $U \subseteq S$ ; $e \in W$ because $e = u \cdot u^{- 1}$ for any $u \in U$ ; and $W \subseteq S \cdot S^{- 1} = S^{2}$ because $U \subseteq S = S^{- 1}$ . For $y \in S^{m}$ the translate $y W$ is open and contains $y = y e$ , while $y W \subseteq S^{m} S^{2} = S^{m + 2}$ ; hence $S^{m} \subseteq int (S^{m + 2})$ for every $m \geq 1$ . Therefore the open sets $int (K_{2 n}) = int (S^{2 n})$ increase, and they cover $G$ : for $x \in S^{m}$ padding with $e \in S$ gives $x \in S^{2 m} \subseteq int (S^{2 m + 2}) = int (K_{2 m + 2})$ , so $x$ lies in the member of index $m + 1$ . If $Q = \emptyset$ , take $n_{0} = 1$ . Otherwise [F4] gives a finite subcover of the compact $Q \subseteq G$ by these ambient open sets; taking the largest index gives $n_{0}$ with $Q \subseteq int (K_{2 n_{0}}) \subseteq K_{2 n_{0}}$ , and step 1.1 gives $Q \subseteq K_{n}$ for all $n \geq 2 n_{0}$ .

2.2A1F1F2step 1.1choose

For each $n \geq 0$ , [F1] applied to the compact set $K_{n + 1}$ with tolerance $1 / (n + 1)$ produces a Borel set $F$ with $0 < μ (F) < \infty$ and $Δ_{K_{n + 1}} (F) \leq 1 / (n + 1)$ , so the family of all such witnesses is nonempty; by [A1] choose a sequence $(F_{n})_{n \geq 0}$ in which $F_{n}$ is a witness for $K_{n + 1}$ at tolerance $1 / (n + 1)$ , so each $F_{n}$ is Borel with $0 < μ (F_{n}) < \infty$ and $Δ_{K_{n + 1}} (F_{n}) \leq 1 / (n + 1)$ .

3.1F2step 2.1step 2.2algebra

Let $Q \subseteq G$ be compact and choose $m$ as in step 2.1, so $Q \subseteq K_{r}$ for all $r \geq 2 m$ . For every $n \geq 2 m - 1$ , one has $Q \subseteq K_{n + 1}$ , so the family defining $Δ_{Q} (F_{n})$ is contained in the family defining $Δ_{K_{n + 1}} (F_{n})$ , hence $0 \leq Δ_{Q} (F_{n}) \leq Δ_{K_{n + 1}} (F_{n}) \leq 1 / (n + 1)$ by step 2.2. This also covers $Q = \emptyset$ because both families include $0$ . Thus $Δ_{Q} (F_{n}) \to 0$ and $(F_{n})$ is a Følner sequence.

4.1A1F1step 3.1step 1.2∎

Steps 1.1, 2.1, 2.2, and 3.1 produce a Følner sequence for an amenable $G$ from the compact sets $S^{n}$ and their open exhaustion of $G$ , and step 1.2 reverses the implication, giving the stated equivalence; the construction never uses second countability, and the stated AC is spent only through the criterion in [F1] and the countable selection in [A1].

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The Hulanicki–Reiter weak containment criterion for amenability

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ . Let $1_{G}$ be the trivial unitary representation on $C$ with its standard inner product, given by $1_{G} (g) z = z$ , and let $λ_{G}$ be the left regular representation on $L^{2} (G)$ (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful). Then $G$ is amenable (Amenable locally compact group) if and only if $1_{G} ≺ λ_{G}$ (Weak containment of unitary representations). Equivalently, $G$ is amenable if and only if $λ_{G}$ almost has invariant unit vectors: for every compact $Q \subseteq G$ and every $ε > 0$ there is a unit vector $ξ \in L^{2} (G)$ with $\sup_{x \in Q} ∥ λ_{G} (x) ξ - ξ ∥_{2} < ε$ .

Facts & Assumptions

Given: AC, a locally compact Hausdorff group $G$ , a fixed left Haar measure $μ$ , and the representations $1_{G}$ on $C$ and $λ_{G}$ on $L^{2} (G)$ .

[A1]

AC is assumed in the choice-function form (The Axiom of Choice).

[F1]

Under AC, amenability of a locally compact Hausdorff group is equivalent to Reiter's condition (P1) (Amenability is equivalent to Reiter's condition (P1)).

[F2]

Reiter (P1) means that for every compact $Q$ and every $δ > 0$ there is $f \in L^{1} (G)$ with $f \geq 0$ , $∥ f ∥_{1} = 1$ , and $\sup_{x \in Q} ∥ L_{x} f - f ∥_{1} \leq δ$ (Reiter's condition (P1)).

[F3]

The left regular action is $λ_{G} (x) ξ (y) = ξ (x^{- 1} y)$ ; it is a unitary representation, and under AC it is strongly continuous (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful).

[F4]

$L^{1} (G)$ and $L^{2} (G)$ are complex almost-everywhere classes with $∥ f ∥_{1} = \int_{G} ∣ f ∣ d μ$ and $∥ ξ ∥_{2}^{2} = \int_{G} ∣ ξ ∣^{2} d μ$ ; the $L^{2}$ pairing is the integral pairing (Complex Haar L^p spaces and compactly supported functions, The complex $L^{2}$ pairing on equivalence classes).

[F5]

For vectors $u, v$ in a complex inner-product space, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F6]

Complex modulus is subadditive and satisfies $∣ ∣ a ∣ - ∣ b ∣ ∣ \leq ∣ a - b ∣$ for $a, b \in C$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F7]

Weak containment $π ≺ ρ$ means that every diagonal coefficient of $π$ is uniformly approximable on compact subsets by finite sums of diagonal coefficients of $ρ$ (Weak containment of unitary representations).

[F8]

Under AC, $1_{G} ≺ π$ is equivalent to existence of unit vectors in $π$ that are arbitrarily invariant on each compact subset (Weak containment of the trivial representation and almost invariant vectors).

[F9]

With the standard inner product on $C$ , the diagonal coefficient of $1_{G}$ at $z \in C$ is the constant function $∣ z ∣^{2}$ ; this follows from $1_{G} (g) z = z$ and the matrix-coefficient definition (Real and complex inner-product spaces and their induced length, Matrix coefficient of a unitary representation).

[F10]

Amenability is existence of a left-invariant mean on $L^{\infty} (G)$ (Amenable locally compact group).

[F11]

A strongly continuous unitary representation has continuous orbit maps; the trivial action on $C$ is constant, hence strongly continuous (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[F12]

Left Haar measure is left invariant, and the left regular action uses $(L_{x} f) (y) = f (x^{- 1} y)$ (Left Haar integral and left Haar measure, Left and right regular unitary representations of an LCH group).

Proof

Given: AC, an LCH group $G$ with fixed left Haar measure $μ$ , and its left regular representation $λ_{G}$ .

Proof technique: direct.

1.1F1F10given

Suppose $G$ is amenable in the sense of [F10]. By [F1], $G$ satisfies Reiter (P1).

1.2F2F3F4F12algebra

Fix compact $Q \subseteq G$ and $ε > 0$ . By [F2] choose $f \in L^{1} (G)$ with $f \geq 0$ , $∥ f ∥_{1} = 1$ , and $\sup_{x \in Q} ∥ L_{x} f - f ∥_{1} \leq ε^{2} / 2$ . Set $g : = \sqrt{f} \in L^{2} (G)$ ; then $∥ g ∥_{2} = 1$ . For $a, b \geq 0$ with $a \neq b$ , $∣ \sqrt{a} - \sqrt{b} ∣^{2} = ∣ a - b ∣ ∣ \sqrt{a} - \sqrt{b} ∣ / (\sqrt{a} + \sqrt{b}) \leq ∣ a - b ∣$ , and for $a = b$ both sides are zero. Thus for every $x \in Q$ , $∥ λ_{G} (x) g - g ∥_{2}^{2} = \int_{G} ∣ \sqrt{f (x^{- 1} y)} - \sqrt{f (y)} ∣^{2} d μ (y) \leq ∥ L_{x} f - f ∥_{1} \leq ε^{2} / 2$ , so $\sup_{x \in Q} ∥ λ_{G} (x) g - g ∥_{2} < ε$ .

1.3F3F5F7F9F11given

Assume that $λ_{G}$ almost has invariant unit vectors. Let $Q$ be compact, $τ > 0$ , and let $z \in C$ with $c : = ∣ z ∣^{2}$ . If $c = 0$ , the coefficient of the zero vector is exactly $0$ . If $c > 0$ , choose a unit vector $ξ \in L^{2} (G)$ with $\sup_{x \in Q} ∥ λ_{G} (x) ξ - ξ ∥_{2} < τ / c$ and put $η : = \sqrt{c} ξ$ . By [F5], $∣ 1 - ⟨ λ_{G} (x) ξ, ξ ⟩ ∣ = ∣ ⟨ ξ - λ_{G} (x) ξ, ξ ⟩ ∣ \leq ∥ λ_{G} (x) ξ - ξ ∥_{2}$ , so $\sup_{x \in Q} ∣ c - ⟨ λ_{G} (x) η, η ⟩ ∣ < τ$ . By [F9] every diagonal coefficient of $1_{G}$ is such a constant $c \geq 0$ , and each approximant used here is one coefficient of $λ_{G}$ ; thus [F7] gives $1_{G} ≺ λ_{G}$ .

1.4A1F3F8given

If $1_{G} ≺ λ_{G}$ , [F8] gives almost invariant unit vectors for $λ_{G}$ on every compact subset.

2.1F2F3F4F5F6F12step 1.4algebra

Fix compact $Q \subseteq G$ and $ε > 0$ . By step 1.4 choose a unit vector $ξ \in L^{2} (G)$ with $\sup_{x \in Q} ∥ λ_{G} (x) ξ - ξ ∥_{2} < ε / 2$ . Set $f : = ∣ ξ ∣^{2} \in L^{1} (G)$ , so $f \geq 0$ and $∥ f ∥_{1} = ∥ ξ ∥_{2}^{2} = 1$ . Since $L_{x} f = ∣ λ_{G} (x) ξ ∣^{2}$ , [F6] gives $∣ ∣ λ_{G} (x) ξ ∣^{2} - ∣ ξ ∣^{2} ∣ \leq ∣ λ_{G} (x) ξ - ξ ∣ (∣ λ_{G} (x) ξ ∣ + ∣ ξ ∣)$ pointwise. By [F5], $∥ L_{x} f - f ∥_{1} \leq ∥ λ_{G} (x) ξ - ξ ∥_{2} ∥ ∣ λ_{G} (x) ξ ∣ + ∣ ξ ∣ ∥_{2}$ ; the pointwise estimate $(∣ a ∣ + ∣ b ∣)^{2} \leq 2 (∣ a ∣^{2} + ∣ b ∣^{2})$ and unitarity [F3] give $∥ ∣ λ_{G} (x) ξ ∣ + ∣ ξ ∣ ∥_{2} \leq 2$ . Therefore $∥ L_{x} f - f ∥_{1} \leq 2 ∥ λ_{G} (x) ξ - ξ ∥_{2} < ε$ for all $x \in Q$ , so Reiter (P1) holds.

3.1A1F1F3F8F10step 1.1step 1.2step 1.3step 1.4step 2.1∎

Reiter (P1) implies amenability in the sense of [F10] by [F1], while steps 1.1–2.1 give amenability implies $1_{G} ≺ λ_{G}$ and $1_{G} ≺ λ_{G}$ implies Reiter (P1). By [F8], weak containment is equivalent to almost invariant unit vectors, proving both formulations in the Statement. AC is used only through the cited suppliers [F1], [F3], and [F8].

Sources

BHV, Kazhdan's Property (T), Appendix G.3, Theorem G.3.2 (Hulanicki–Reiter), printed pp. 456–457, proves amenability iff $1_{G} ≺ λ_{G}$ by converting between Reiter densities and almost-invariant $L^{2}$ vectors. Appendix F.1, Corollary F.1.5 and its proof, printed pp. 423–424, gives the weak-containment/almost-invariant-vector equivalence. Thomas, Lecture 20, slides 17–18 (PDF pp. 17–18), gives the same Reiter-to- $L^{2}$ conversion.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The Markov-Kakutani fixed point theorem for abelian affine actions

Statement

Assume the Axiom of Choice. Let $G$ be an abelian topological group, and let $X$ be a nonempty compact convex subset of a Hausdorff locally convex real or complex topological vector space $V$ . Suppose $G$ acts continuously on $X$ ; write $g x$ for the action, with $e x = x$ and $(g h) x = g (h x)$ . Each action map is affine in the finite-combination sense: for $x_{0}, \dots, x_{n} \in X$ and real $t_{i} \geq 0$ with $\sum_{i = 0}^{n} t_{i} = 1$ , $g (\sum_{i = 0}^{n} t_{i} x_{i}) = \sum_{i = 0}^{n} t_{i} (g x_{i}) .$ Then there exists $x_{0} \in X$ with $g x_{0} = x_{0}$ for every $g \in G$ .

Facts & Assumptions

Given: The Axiom of Choice, an abelian topological group $G$ , a Hausdorff locally convex topological vector space $V$ , a nonempty compact convex subset $X \subseteq V$ , and the continuous affine action in the Statement.

[A1]

Under the Axiom of Choice, the real dominated-extension theorem proves the relative Hahn-Banach principle HB (The Axiom of Choice, Hahn-Banach dominated extension theorem for real vector spaces, The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

Addition and scalar multiplication in $V$ are continuous; convexity is defined by finite convex combinations (Topological vector spaces over the real and complex fields, Local convexity, convex and balanced sets, and the continuous dual).

[F2]

A continuous image of a compact space is compact; $X$ is Hausdorff as a subspace of the Hausdorff space $V$ , so a compact subset of $X$ is closed in $X$ ; and a family of closed subsets of compact $X$ with the finite-intersection property has nonempty intersection (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, A space is compact exactly when every family of closed subsets with the finite intersection property has nonempty intersection).

[F3]

In a Hausdorff locally convex space, HB implies that the continuous dual separates distinct points by the real part of a functional (The continuous dual separates points in a Hausdorff locally convex space).

[F4]

A continuous real-valued function on a nonempty compact space is bounded and attains its extrema (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F5]

For a vector sequence, define its finite sums recursively by $S_{0} = v_{0}$ and $S_{k + 1} = S_{k} + v_{k + 1}$ ; vector-space axioms and induction give distributivity and reindexing of finite sums. Real finite sums obey additivity, scaling, and telescoping. The canonical natural $n + 1$ is positive, its reciprocal is positive, and reciprocals decrease as positive denominators increase (Vector space over a field, The recursion theorem, The principle of mathematical induction, Finite sums and finite products, by recursion, Laws of finite sums and finite products, The natural numbers $N$ (von Neumann), Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order).

[F6]

For every real $ε > 0$ there is a natural $n \geq 1$ with $1 / n < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

Proof

technique · direct

1.1F1F5given

For $g \in G$ and $n \in N$ , define $g^{0} x = x$ , $g^{i + 1} x = g (g^{i} x)$ , and $A_{n} (g) x : = \frac{1}{n + 1} \sum_{i = 0}^{n} g^{i} x$ , with vector-valued finite sums as in [F5]. Since $1 / (n + 1) > 0$ and the sum of the $n + 1$ equal coefficients is $(n + 1) / (n + 1) = 1$ , convexity makes $A_{n} (g)$ a self-map of $X$ . The action iterates are continuous and affine by induction. For any finite convex combination $y = \sum_{ℓ} t_{ℓ} y_{ℓ}$ , their affine identities and finite sum distributivity give $A_{n} (g) y = \frac{1}{n + 1} \sum_{i} \sum_{ℓ} t_{ℓ} g^{i} y_{ℓ} = \sum_{ℓ} t_{ℓ} A_{n} (g) y_{ℓ}$ ; hence $A_{n} (g)$ is affine. Continuity follows from the TVS addition and scalar-multiplication maps and the continuity of the action iterates.

2.1F1F5step 1.1givenalgebra

Let $Γ$ be the monoid of finite compositions of the maps $A_{n} (g)$ , including the identity. For $g, h \in G$ , affinity and the action law give $A_{n} (g) A_{m} (h) x = \frac{1}{(n + 1) (m + 1)} \sum_{i = 0}^{n} \sum_{j = 0}^{m} g^{i} h^{j} x$ and $A_{m} (h) A_{n} (g) x = \frac{1}{(n + 1) (m + 1)} \sum_{j = 0}^{m} \sum_{i = 0}^{n} h^{j} g^{i} x$ . Because $G$ is abelian, $g^{i} h^{j} = h^{j} g^{i}$ ; associativity and commutativity of vector addition and scalar distributivity reorder the finite sums, so the two maps commute. Therefore $Γ$ is abelian, and every member is a continuous self-map of $X$ .

3.1F2step 2.1

For every $γ \in Γ$ , the image $γ (X)$ is nonempty and compact by [F2], hence closed in $X$ by [F2]. Given a nonempty finite list $γ_{1}, \dots, γ_{k} \in Γ$ , their composition $γ : = γ_{1} \dots γ_{k}$ lies in $Γ$ ; commutativity lets us write $γ = γ_{i} \circ γ_{i}^{'}$ for each $i$ with $γ_{i}^{'}$ the composition of the other factors, using the identity when $k = 1$ . Thus the nonempty set $γ (X)$ lies in every $γ_{i} (X)$ . The empty finite intersection is $X \neq \emptyset$ , so ${γ (X) : γ \in Γ}$ has the finite intersection property, and compactness of $X$ gives $x_{0} \in ⋂_{γ \in Γ} γ (X)$ .

4.1A1F3F4step 3.1

Fix $g \in G$ and suppose $v : = x_{0} - g x_{0} \neq 0$ . By [A1] and [F3], choose a continuous linear functional $φ \in V^{'}$ whose real part $ψ : = Re φ$ satisfies $ψ (v) \neq 0$ . The continuous real-valued function $ψ$ is bounded on compact $X$ by [F4]; fix $C \geq 0$ with $∣ ψ (y) ∣ \leq C$ for every $y \in X$ .

5.1F5F6step 4.1algebra∎

For every $n \in N$ , membership $x_{0} \in A_{n} (g) (X)$ gives some $x \in X$ with $x_{0} = A_{n} (g) x$ . Affinity of the action and real-linearity of $ψ$ yield $ψ (v) = (ψ (x) - ψ (g^{n + 1} x)) / (n + 1)$ by telescoping, so $∣ ψ (v) ∣ \leq 2 C / (n + 1)$ . This bound is valid for every $n$ ; no sequence of preimages is chosen. If $C = 0$ , the bound gives $ψ (v) = 0$ directly. If $C > 0$ , then for any $ε > 0$ , [F6] applied to $ε / (2 C) > 0$ gives $n \geq 1$ with $1 / n < ε / (2 C)$ . Since $n + 1 > n > 0$ , [F5] gives $1 / (n + 1) < 1 / n$ , hence $∣ ψ (v) ∣ \leq 2 C / (n + 1) < ε$ . As this holds for every positive $ε$ , $ψ (v) = 0$ , contradicting step 4.1. Therefore $g x_{0} = x_{0}$ . Since $g$ was arbitrary, $x_{0}$ is fixed by all of $G$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The fixed point property implies amenability

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with the fixed point property: every continuous affine action of $G$ on a nonempty compact convex subset of a Hausdorff locally convex topological vector space has a fixed point. Then $G$ is amenable (Amenable locally compact group).

Facts & Assumptions

Given: AC, an LCH group $G$ , and the fixed point property in the Statement.

[A1]

AC is assumed in the choice-function form (The Axiom of Choice).

[F1]

$X : = U C B (G)$ consists of actual bounded continuous functions with the supremum norm; it is invariant under left translations, and the orbit map $x \mapsto L_{x} ψ$ is norm-continuous for each $ψ \in X$ (Left-uniformly continuous bounded functions (UCB)).

[F2]

Complex conjugation, real and imaginary parts, and modulus have their coordinate definitions and standard modulus laws (Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F3]

The continuous dual $X^{*}$ consists of bounded linear functionals with the dual norm; the weak-star topology is the initial topology of the evaluation maps $m \mapsto m (ψ)$ and has a finite-evaluation neighborhood basis (The dual space X^* of a normed space and its dual norm, The weak-star topology from finite evaluations).

[F4]

A topological vector space has jointly continuous addition and scalar multiplication; local convexity means that zero has a base of convex neighborhoods (Topological vector spaces over the real and complex fields, Local convexity, convex and balanced sets, and the continuous dual).

[F5]

AC implies that every filter extends to an ultrafilter (The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter).

[F6]

Under the ultrafilter lemma, the closed dual unit ball of a normed space is weak-star compact (Banach–Alaoglu).

[F7]

A closed subspace of a compact space is compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[F8]

A left-invariant mean on $U C B (G)$ yields Reiter's condition (P1) (An invariant mean produces a Reiter net).

[F9]

Under the ultrafilter lemma, a Reiter net has a weak-star cluster point which is a left-invariant mean on $L^{\infty} (G)$ (A Reiter net has an invariant-mean cluster point).

[F10]

Amenability means existence of a left-invariant mean on complex $L^{\infty} (G)$ (Amenable locally compact group, Left-invariant means on $L^{\infty}$ of a locally compact group).

[F12]

Reiter's condition (P1) is equivalent to the existence of a net in $P$ with the compact-uniform translation estimates (Reiter's condition (P1)).

Proof

technique · direct

1.1F3F4

Put $X : = U C B (G)$ and $E : = X^{*}$ with the weak-star topology. By [F3], every evaluation on $E$ is continuous and linear. Therefore addition and scalar multiplication on $E$ are continuous, since their evaluations are the corresponding sums and scalar multiples. The basic zero-neighborhoods are finite intersections of inverse images of open disks under linear evaluations; these neighborhoods are convex. Distinct functionals differ on some $ψ \in X$ ; disjoint scalar neighborhoods of their evaluations pull back to disjoint weak-star neighborhoods, so $E$ is Hausdorff. Hence $E$ is a Hausdorff locally convex topological vector space by [F4].

1.2F1F2F3algebra

Let $M$ be the set of positive complex-linear functionals $m$ on $X$ with $m (1_{G}) = 1$ . It is nonempty because evaluation $δ_{e} (ψ) = ψ (e)$ is a mean, and it is convex. Every real-valued $u \in X$ satisfies $m (u) \in R$ and $∣ m (u) ∣ \leq ∥ u ∥_{\infty}$ : positivity applied to $∥ u ∥_{\infty} 1_{G} + u$ and $∥ u ∥_{\infty} 1_{G} - u$ gives both claims. Real and imaginary parts of UCB functions remain in $X$ , since their translation differences are bounded by the original difference. If $m (ψ) \neq 0$ , set $α = \overline{m (ψ)} / ∣ m (ψ) ∣$ . Then $∣ α ∣ = 1$ and $m (Re (α ψ)) = ∣ m (ψ) ∣$ ; also $Re (α ψ) \leq ∣ ψ ∣ \leq ∥ ψ ∥_{\infty} 1_{G}$ . Positivity gives $∣ m (ψ) ∣ \leq ∥ ψ ∥_{\infty}$ , and the same bound is immediate if $m (ψ) = 0$ . Thus $M \subseteq B_{X^{*}}$ .

2.1A1F3F5F6F7step 1.1step 1.2

The set $M$ is weak-star closed: it is the intersection of ${m : m (1_{G}) = 1}$ and, for every nonnegative $ψ \in X$ , ${m : m (ψ) \in [0, \infty)}$ ; these are closed by [F3]. By [A1] and [F5], the ultrafilter lemma holds, so [F6] makes $B_{X^{*}}$ compact. Since $M$ is a closed subset, [F7] makes $M$ compact. Together with step 1.2, $M$ is a nonempty compact convex subset of the locally convex space $E$ .

2.2F1F3step 1.2

For $g \in G$ and $m \in M$ , define $(g \cdot m) (ψ) : = m (L_{g^{- 1}} ψ)$ for $ψ \in X$ . Translation invariance of $X$ shows this is well-defined; positivity and $L_{g^{- 1}} 1_{G} = 1_{G}$ show $g \cdot m \in M$ . The identity $L_{a} L_{b} = L_{a b}$ gives $g \cdot (h \cdot m) = (g h) \cdot m$ , and linearity in $m$ makes each map $m \mapsto g \cdot m$ affine.

3.1F1F3step 1.2step 2.2

Fix $(g_{0}, m_{0}) \in G \times M$ , $ψ \in X$ , and $η > 0$ . By [F1], choose a neighborhood $V$ of $g_{0}$ with $∥ L_{g^{- 1}} ψ - L_{g_{0}^{- 1}} ψ ∥_{\infty} < η / 2$ for $g \in V$ . By [F3], choose a weak-star neighborhood $W$ of $m_{0}$ such that $∣ (m - m_{0}) (L_{g_{0}^{- 1}} ψ) ∣ < η / 2$ for $m \in W$ . For $g \in V$ and $m \in W \cap M$ , step 1.2 gives $∣ (g \cdot m) (ψ) - (g_{0} \cdot m_{0}) (ψ) ∣ \leq ∥ m ∥ ∥ L_{g^{- 1}} ψ - L_{g_{0}^{- 1}} ψ ∥_{\infty} + ∣ (m - m_{0}) (L_{g_{0}^{- 1}} ψ) ∣ < η$ . Thus every evaluation of the action is continuous; by the initial weak-star topology, the action $G \times M \to M$ is continuous. It is affine by step 2.2.

4.1F1step 2.1step 2.2step 3.1given

The fixed point property applied to the continuous affine action of step 2.2 on the nonempty compact convex set $M$ gives a fixed point $m \in M$ . Thus $m (L_{g^{- 1}} ψ) = m (ψ)$ for every $g \in G$ and $ψ \in X$ ; as $g^{- 1}$ ranges over $G$ , $m$ is a left-invariant mean on $U C B (G)$ .

5.1A1F5F8F9F10F12step 4.1∎

By [F8] and step 4.1, $G$ satisfies (P1); [F12] gives a Reiter net. AC supplies the ultrafilter lemma by [F5], so [F9] gives a left-invariant mean on $L^{\infty} (G)$ . By [F10], $G$ is amenable.

Sources

BHV, Kazhdan's Property (T), Appendix G.1, Remark G.1.6 and Theorem G.1.7, proves that the fixed-point property for continuous affine actions on nonempty compact convex sets in locally convex spaces implies amenability, using the weak-star compact state space of UCB means and its translation action (printed pp. 448–449). The proof above supplies the compactness and continuity details under the repository's explicit AC convention, then uses the local Reiter and cluster-point suppliers to reach the stated $L^{\infty}$ definition.

PropositionStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Compact and locally compact abelian groups are amenable

Statement

Assume AC. (1) Every compact locally compact Hausdorff group is amenable. (2) Every locally compact Hausdorff abelian group is amenable. No countability, metrizability or unimodularity hypothesis is imposed.

Facts & Assumptions

Given: AC, a compact locally compact Hausdorff group $K$ in part (1), and an LCH abelian group $G$ in part (2).

[A1]

AC is assumed in the choice-function form (The Axiom of Choice).

[F1]

Under AC, a locally compact Hausdorff group has a left Haar measure $ν$ (Existence of left and right Haar measures, Left Haar integral and left Haar measure). A compact group $K$ is open in itself and compact, so its Haar measure satisfies $0 < ν (K) < \infty$ (Haar measure is positive on nonempty open sets and finite on compact sets). The rescaled measure $μ (E) : = ν (E) / ν (K)$ is again left Haar and has $μ (K) = 1$ .

[F2]

Complex $L^{\infty}$ consists of Borel almost-everywhere classes with the essential-supremum norm; integrability means finiteness of the integral of the modulus, and integrals of integrable functions respect almost-everywhere equality (Complex $L^{\infty}$ space of a locally compact group, Integrable real and complex functions, and their integrals, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree).

[F3]

The complex integral is linear and satisfies the integral triangle inequality; the nonnegative integral is monotone and respects nonnegative scalars, with the integral of an indicator equal to the measure of its set (The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, The nonnegative Lebesgue integral, The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions).

[F4]

Left translations are Borel measure-preserving maps for left Haar measure, and integrals are invariant under measure-preserving maps (Left Haar integral and left Haar measure, A continuous map has Borel preimages of Borel sets, Measure-preserving transformations and systems, Integral invariance under measure-preserving maps).

[F5]

A mean on complex $L^{\infty} (G)$ is positive, complex-linear and unital; left invariance means $m (L_{g} φ) = m (φ)$ for all $g$ and $φ$ (Left-invariant means on $L^{\infty}$ of a locally compact group).

[F6]

Amenability means existence of such a left-invariant mean (Amenable locally compact group).

[F7]

Under AC, every continuous affine action of an abelian topological group on a nonempty compact convex subset of a Hausdorff locally convex space has a fixed point (The Markov-Kakutani fixed point theorem for abelian affine actions).

[F8]

Under AC, the fixed-point property implies amenability (The fixed point property implies amenability).

Proof

technique · direct

1.1A1F1F2F3F5construct

Let $K$ be compact. By [A1, F1], choose a left Haar measure $ν$ and set $μ : = ν / ν (K)$ ; [F1] gives $0 < ν (K) < \infty$ , and positive scalar rescaling preserves left invariance and regularity, so $μ$ is a normalized Haar probability. For $φ \in L^{\infty} (K, μ; C)$ and every $η > 0$ , the definition of essential supremum gives $∣ φ ∣ \leq ∥ φ ∥_{\infty} + η$ almost everywhere. Since $μ (K) = 1$ , monotonicity of the nonnegative integral gives $\int_{K} ∣ φ ∣ d μ \leq ∥ φ ∥_{\infty} + η < \infty$ , so $φ$ is integrable by [F2]. Define $m (φ) : = \int_{K} φ d μ$ . Its definition is independent of the representative by [F2]; linearity and positivity follow from [F3], while the simple-function integral gives $m (1_{K}) = μ (K) = 1$ . The triangle inequality gives $∣ m (φ) ∣ \leq ∥ φ ∥_{\infty} + η$ for every $η > 0$ , hence $∣ m (φ) ∣ \leq ∥ φ ∥_{\infty}$ . Thus $m$ is a bounded positive unital functional, hence a mean by [F5].

1.2A1F7F8

Let $G$ be locally compact Hausdorff and abelian. By [F7], every continuous affine action of $G$ on a nonempty compact convex subset of a Hausdorff locally convex space has a fixed point. Thus $G$ has the fixed-point property required by [F8]. Applying [F8] proves that $G$ is amenable.

2.1F4F5F6step 1.1

For $g \in K$ , the left translation $T_{g} (x) = g^{- 1} x$ is Borel and measure-preserving by [F4]. It therefore preserves null sets, so composition defines the same $L^{\infty}$ class independently of the representative. The integral invariance in [F4] gives $m (L_{g} φ) = \int_{K} φ \circ T_{g} d μ = \int_{K} φ d μ = m (φ)$ for every $φ \in L^{\infty} (K)$ . Thus the mean from step 1.1 is left-invariant, and $K$ is amenable by [F5, F6].

3.1step 1.2step 2.1∎

Step 2.1 proves part (1), and step 1.2 proves part (2), with no countability or unimodularity assumption.

Sources

BHV, Kazhdan's Property (T), Appendix G.1, Example G.1.5 and Theorem G.1.7, and Appendix G.2, Theorem G.2.1, printed pp. 448–451. The local proof constructs the compact-group mean by integration and uses the local Markov–Kakutani and fixed-point-property lemmas for the abelian case.

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-10-08Open item page →

Restriction of the regular representation to a closed subgroup

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group and let $H \leq G$ be closed. Fix left Haar measures on $G$ and $H$ . Then the restriction of the left regular representation of $G$ to $H$ is weakly contained in the left regular representation of $H$ : $λ_{G} ∣_{H} ≺ λ_{H},$ where weak containment means uniform approximation of each diagonal coefficient on every compact subset of $H$ by finite sums of diagonal coefficients.

Facts & Assumptions

Given: AC, a locally compact Hausdorff group $G$ , a closed subgroup $H$ , and fixed left Haar measures $d x$ on $G$ and $d h$ on $H$ .

[A1]

AC is the choice-function principle (The Axiom of Choice).

[F1]

There is a positive continuous rho-function $r$ for $(G, H)$ and a Radon measure $μ$ on $G / H$ such that $\int_{G} u (y) r (y) d y = \int_{G / H} \int_{H} u (x h) d h d μ (x H)$ for every $u \in C_{c} (G)$ ; by real and imaginary parts it also holds for complex $u$ (Existence of rho-functions and quotient measure classes, Weil formula with a rho-function).

[F2]

The closed subspace $H$ is locally compact and Hausdorff; $G / H$ is locally compact Hausdorff and the quotient map $p : G \to G / H$ is open. Locally compact Hausdorff spaces have compact neighborhoods (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, In a locally compact Hausdorff space every point has a neighbourhood base of compact sets, every open subspace and every closed subspace is locally compact, every open set around a point contains an open set with compact closure inside it, and every compact set sits inside an open set with compact closure, Compact lifts and averaging onto C_c(G/H)).

[F3]

The modular functions are positive continuous homomorphisms and $r (x h) = \frac{Δ_{H} (h)}{Δ_{G} (h)} r (x)$ (Rho-function for a closed subgroup, The modular function is a continuous homomorphism).

[F4]

Inversion changes left Haar integration by $\int_{G} u (y^{- 1}) d y = \int_{G} u (y) Δ_{G} (y^{- 1}) d y$ for nonnegative Borel $u$ and for complex Borel $u$ satisfying $\int_{G} Δ_{G} (y^{- 1}) ∣ u (y) ∣ d y < \infty$ . The same identity applies on $H$ with $Δ_{H}$ (Haar change of variables under inversion).

[F5]

A compactly supported continuous kernel on a product of locally compact Hausdorff spaces has continuous compactly supported partial integrals; the two positive Radon integrations commute, and the result extends to complex kernels (Compactly supported kernels admit commuting radon integrals).

[F6]

On complex $L^{2}$ the left and right regular representations are strongly continuous and unitary, with $λ_{G} (k) f (x) = f (k^{- 1} x), R_{H} (k) b (h) = Δ_{H} (k)^{1 / 2} b (h k) .$ Also $C_{c} (G)$ and $C_{c} (H)$ are the continuous complex functions of compact support, are dense in their respective $L^{2}$ spaces, and those spaces are complete (Compact support, $C_{c} (X)$ , and $C_{0} (X)$ , Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful, Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F7]

For every $ξ \in L^{2} (G)$ , compact $Q \subseteq H$ and $ϵ > 0$ , $λ_{G} ∣_{H} ≺ λ_{H}$ means that there are finitely many $η_{j} \in L^{2} (H)$ with $\sup_{k \in Q} ∣ ⟨ λ_{G} (k) ξ, ξ ⟩ - \sum_{j} ⟨ λ_{H} (k) η_{j}, η_{j} ⟩ ∣ < ϵ$ (Weak containment of unitary representations).

[F8]

In an inner-product space, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F9]

Continuous images of compact sets and closed subsets of compact spaces are compact; compact subsets of Hausdorff spaces are closed; open sets generate the Borel sigma-algebra, which is closed under finite unions, intersections and complements (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, The Borel sigma-algebra of a topological space).

[F10]

Every nonnegative real number has a nonnegative square root (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ).

[F11]

Proof

technique · direct

1.1A1F1F3F6F9construct

Fix $f \in C_{c} (G)$ and put $S = supp f$ . For $x \in G$ define $a_{x} \in C_{c} (H)$ by $a_{x} (h) : = {(\frac{Δ_{G} ((x h)^{- 1})}{r (x h)})}^{1 / 2} f ((x h)^{- 1})$ . Its support is contained in the compact set $x^{- 1} S^{- 1} \cap H$ . For $k \in H$ put $B (x H, k) : = ⟨ R_{H} (k) a_{x}, a_{x} ⟩$ . If $t \in H$ , then $a_{x t} (h) = a_{x} (t h)$ ; changing $h$ by the left translation $t$ in the Haar integral shows $B (x t H, k) = B (x H, k)$ . Thus $B$ is well defined on $(G / H) \times H$ .

2.1F2F5F9step 1.1construct

The function $(x, k, h) ⟼ Δ_{H} (k)^{1 / 2} a_{x} (h k) \overline{a_{x} (h)}$ is continuous. At each $(x_{0}, k_{0}) \in G \times H$ , choose compact neighborhoods $C ∋ x_{0}$ and $K ∋ k_{0}$ by [F2]. For $x \in C$ and $k \in K$ , its support in $h$ lies in the fixed compact set $D : = C^{- 1} S^{- 1} \cap H$ , because the second factor vanishes unless $x h \in S^{- 1}$ ; compactness of $D$ follows from [F9]. The support of the restricted kernel on $C \times K \times H$ is contained in the compact product $C \times K \times D$ , which is compact by A product of finitely many compact spaces is compact in the product topology. The compact-kernel result [F5] therefore makes the integral over $h$ continuous jointly in $(x, k)$ near $(x_{0}, k_{0})$ . The map $p \times {id}_{H}$ is open on product-basis rectangles and surjective, hence is a quotient map; since $B$ is constant on its fibers, it descends continuously. If $q \notin T : = p (S^{- 1})$ , no representative $x$ has $x h \in S^{- 1}$ , so $a_{x} = 0$ and $B (q, k) = 0$ . The set $T$ is compact by continuity of $p$ and [F9].

3.1F1F3F4F6F9step 1.1step 2.1algebra

The coefficient $c_{f} (k) : = ⟨ λ_{G} (k) f, f ⟩$ has the formula $c_{f} (k) = \int_{G / H} B (q, k) d μ (q)$ . Indeed, inversion [F4] gives $c_{f} (k) = \int_{G} Δ_{G} (y^{- 1}) f ((y k)^{- 1}) \overline{f (y^{- 1})} d y$ . The inversion input is in $C_{c} (G)$ , so its weighted absolute integral is finite by continuity of $Δ_{G}$ and compact finiteness of Haar measure. The resulting integrand is continuous and supported in the compact set $S^{- 1} \cap S^{- 1} k^{- 1}$ . Apply the Weil formula [F1] after dividing it by $r (y)$ ; for a complex integrand apply the real formula to its real and imaginary parts. At $y = x h$ the resulting integrand is $\frac{Δ_{G} ((x h)^{- 1})}{r (x h)} f ((x h k)^{- 1}) \overline{f ((x h)^{- 1})}$ . The rho covariance [F3] gives $Δ_{H} (k)^{1 / 2} a_{x} (h k) \overline{a_{x} (h)} = \frac{Δ_{G} ((x h)^{- 1})}{r (x h)} f ((x h k)^{- 1}) \overline{f ((x h)^{- 1})}$ , which is exactly the inner product defining $B (x H, k)$ . Since $B$ vanishes off compact $T$ and $μ (T) < \infty$ , this quotient integral is finite.

4.1A1F1F7F9F10F11step 2.1step 3.1choose

Fix a compact $Q \subseteq H$ and $ϵ > 0$ . If $Q = \emptyset$ , the approximation condition is vacuous, so take one zero vector. If $μ (T) = 0$ , the coefficient formula is zero on $Q$ , so again take one zero vector. Otherwise $μ (T) > 0$ . Set $δ : = ϵ / (2 μ (T)) > 0$ . For each $q_{0} \in T$ , joint continuity of $B$ gives, at each $k_{0} \in Q$ , neighborhoods $U_{k_{0}}$ of $q_{0}$ and $V_{k_{0}}$ of $k_{0}$ such that both $B (q^{'}, k)$ and $B (q^{''}, k)$ lie within $δ / 2$ of $B (q_{0}, k_{0})$ whenever $q^{'}, q^{''} \in U_{k_{0}} \cap T$ and $k \in V_{k_{0}}$ . By [F11], compactness of $Q$ gives finitely many $V_{k_{0}}$ covering it; intersect their corresponding $U_{k_{0}}$ to obtain a neighborhood $U_{q_{0}}$ on which $∣ B (q^{'}, k) - B (q^{''}, k) ∣ < δ$ for all $q^{'}, q^{''} \in U_{q_{0}} \cap T$ and $k \in Q$ . By [F11], compactness of $T$ gives a finite subcover $U_{1}, \dots, U_{N}$ . The compact set $T$ is closed in the Hausdorff space $G / H$ by [F9]; form the disjoint Borel partition $E_{j} = (U_{j} \cap T) ∖ ⋃_{i < j} U_{i}$ , omitting empty pieces. It covers $T$ , and each $E_{j} \subseteq U_{j}$ . Choose $q_{j} \in E_{j}$ and a lift $x_{j} \in G$ with $p (x_{j}) = q_{j}$ . Radon finiteness gives $μ (E_{j}) < \infty$ . The square root in $v_{j} : = μ (E_{j})^{1 / 2} a_{x_{j}}$ exists by [F10], and $⟨ R_{H} (k) v_{j}, v_{j} ⟩ = μ (E_{j}) B (q_{j}, k)$ . Since the $E_{j}$ partition $T$ and $B$ vanishes off $T$ , $c_{f} (k) = \sum_{j} \int_{E_{j}} B (q, k) d μ (q)$ . Comparing each integral with $μ (E_{j}) B (q_{j}, k)$ gives $\sup_{k \in Q} ∣ c_{f} (k) - \sum_{j} ⟨ R_{H} (k) v_{j}, v_{j} ⟩ ∣ \leq δ μ (T) = ϵ / 2 < ϵ$ . Thus every $C_{c} (G)$ diagonal coefficient is uniformly approximated on $Q$ by a finite sum of right-regular diagonal coefficients.

5.1F3F4F6step 4.1algebra

Define $J : C_{c} (H) \to C_{c} (H)$ by $(J b) (h) : = Δ_{H} (h)^{- 1 / 2} b (h^{- 1})$ . The function $J b$ is continuous with compact support because inversion is a homeomorphism and $Δ_{H}$ is positive continuous. Applying inversion [F4] to $u (h) = Δ_{H} (h) ∣ b (h) ∣^{2}$ gives $∥ J b ∥_{2}^{2} = \int_{H} Δ_{H} (h)^{- 1} ∣ b (h^{- 1}) ∣^{2} d h = \int_{H} ∣ b (h) ∣^{2} d h = ∥ b ∥_{2}^{2}$ . The homomorphism law in [F3] gives $J^{2} b (h) = Δ_{H} (h)^{- 1 / 2} Δ_{H} (h^{- 1})^{- 1 / 2} b (h) = b (h)$ . Also $J λ_{H} (k) b (h) = Δ_{H} (h)^{- 1 / 2} b ((h k)^{- 1})$ , while $R_{H} (k) J b (h) = Δ_{H} (k)^{1 / 2} Δ_{H} (h k)^{- 1 / 2} b ((h k)^{- 1}) = Δ_{H} (h)^{- 1 / 2} b ((h k)^{- 1})$ , so $J λ_{H} (k) = R_{H} (k) J$ . By density and completeness in [F6], $J$ extends to an isometry on $L^{2} (H)$ ; $J^{2} = I$ makes it onto, hence unitary. Thus $⟨ R_{H} (k) v_{j}, v_{j} ⟩ = ⟨ λ_{H} (k) J v_{j}, J v_{j} ⟩$ , and replacing every $v_{j}$ in step 4.1 by $J v_{j}$ converts its sum to left-regular coefficients.

6.1F6F7F8step 4.1step 5.1given∎

Now let $ξ \in L^{2} (G)$ , compact $Q \subseteq H$ , and $ϵ > 0$ . Set $α : = \min {1, ϵ / (8 (∥ ξ ∥_{2} + 1))} > 0$ . By $C_{c} (G)$ -density [F6] choose $f \in C_{c} (G)$ with $∥ ξ - f ∥_{2} < α$ . Then $(∥ ξ ∥_{2} + ∥ f ∥_{2}) ∥ ξ - f ∥_{2} \leq (2 ∥ ξ ∥_{2} + α) α < ϵ / 2$ . For every $k \in H$ , unitarity and Cauchy--Schwarz [F8] give $∣ ⟨ λ_{G} (k) ξ, ξ ⟩ - ⟨ λ_{G} (k) f, f ⟩ ∣ \leq (∥ ξ ∥_{2} + ∥ f ∥_{2}) ∥ ξ - f ∥_{2} < ϵ / 2$ . Apply steps 4.1 and 5.1 to $f$ , $Q$ , and tolerance $ϵ / 2$ , and combine the two bounds. The resulting finite sum of $λ_{H}$ diagonal coefficients approximates the coefficient of $ξ$ within $ϵ$ uniformly on $Q$ . By [F7] this is $λ_{G} ∣_{H} ≺ λ_{H}$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Amenability is stable under closed subgroups, quotients and extensions

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group. (i) If $G$ is amenable, every closed subgroup $H \leq G$ is amenable. (ii) If $G$ is amenable and $N ⊴ G$ is closed normal, the Hausdorff quotient $G / N$ is amenable. (iii) If $N ⊴ G$ is closed normal and both $N$ and $G / N$ are amenable, then $G$ is amenable.

Facts & Assumptions

Given: AC, an LCH group $G$ , and the closed subgroup $H$ or closed normal subgroup $N$ appearing in each clause.

[A1]

AC is the choice-function principle (The Axiom of Choice).

[F1]

Every LCH group has a left Haar measure under AC; this applies to $G$ , a closed subgroup, and the closed-normal quotient once its LCH property is established (Existence of left and right Haar measures).

[F2]

A closed subspace of an LCH space is locally compact, and a subspace of a Hausdorff space is Hausdorff; subgroup operations inherit continuity from the ambient topological group, and the inclusion of a subspace is continuous (In a locally compact Hausdorff space every point has a neighbourhood base of compact sets, every open subspace and every closed subspace is locally compact, every open set around a point contains an open set with compact closure inside it, and every compact set sits inside an open set with compact closure, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Continuity of a map of topological spaces at a point and globally, Topological group: multiplication and inversion are continuous).

[F3]

For closed $N ⊴ G$ , the canonical projection $p : G \to G / N$ is open and the quotient is locally compact Hausdorff; every compact quotient subset has a compact lift (Compact lifts and averaging onto C_c(G/H)).

[F4]

The set of left cosets $G / N$ has the quotient group law and $p$ is a surjective homomorphism (Normal subgroup: invariance under conjugation, The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ , For $N ⊴ G$ , the cosets form a group with identity $N$ and inverse $(g N)^{- 1} = g^{- 1} N$ ).

[F5]

In a topological group multiplication and inversion are continuous, with the product topology on the square (Topological group: multiplication and inversion are continuous, The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

[F6]

$U C B (G)$ consists of actual bounded continuous functions, is translation invariant, and embeds isometrically into complex $L^{\infty}$ by the class map (Left-uniformly continuous bounded functions (UCB)).

[F7]

Amenability gives a positive complex-linear unital invariant mean on $L^{\infty}$ ; such a mean has norm one. Restricting along the isometric UCB class map gives a positive unital invariant mean on UCB, bounded by the sup norm (Amenable locally compact group, Left-invariant means on $L^{\infty}$ of a locally compact group, [F6]).

[F8]

Under AC, a left-invariant mean on UCB gives Reiter (P1), and Reiter (P1) implies amenability (An invariant mean produces a Reiter net, Amenability is equivalent to Reiter's condition (P1)).

[F9]

Under AC and for a fixed left Haar measure, amenability of an LCH group $K$ is equivalent to $1_{K} ≺ λ_{K}$ (The Hulanicki–Reiter weak containment criterion for amenability).

[F10]

Weak containment means uniform approximation of each diagonal coefficient on compact sets by finite sums of diagonal coefficients; this relation is transitive by approximating each of finitely many intermediate coefficients with error divided by their number (Weak containment of unitary representations).

[F11]

If $H$ is closed in $G$ , then the restriction of the left regular representation of $G$ to $H$ is weakly contained in the left regular representation of $H$ (Restriction of the regular representation to a closed subgroup).

[F12]

Under AC the left regular representation is a strongly continuous unitary representation; restricting its parameter to a subgroup with the subspace topology preserves these properties by composition with the continuous inclusion (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Continuity of a map of topological spaces at a point and globally).

[F13]

[F14]

A quotient map is continuous and surjective, and a map out of its target is continuous exactly when its composite with the quotient map is. A continuous open surjection is a quotient map (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps, For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map).

[F15]

The product topology has a basis of open rectangles; a map into a product is continuous when its coordinate maps are continuous (The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

Proof

technique · direct

1.1F3F4F5F14F15construct

Fix a closed normal $N ⊴ G$ and let $p : G \to K : = G / N$ be the canonical projection with quotient topology. By [F3], $K$ is LCH Hausdorff and $p$ is open. The map $p$ is continuous and surjective by [F14]. The product map $p \times p : G \times G \to K \times K$ is continuous by the rectangle basis in [F15], is surjective since each of two cosets has a representative, and is open: every open subset of $G \times G$ is a union of open rectangles $U \times V$ , whose images are $p (U) \times p (V)$ and are open. Thus $p \times p$ is a quotient map by [F14]. The quotient group law [F4] gives $p \circ ι_{G} = ι_{K} \circ p$ and $p \circ m_{G} = m_{K} \circ (p \times p)$ . Since $G$ is a topological group [F5], both left-hand composites are continuous. The quotient-map continuity test [F14] therefore makes inversion $ι_{K}$ and multiplication $m_{K}$ continuous. Hence $K$ is a topological group.

1.2A1F1F2F9F10F11F12F13construct

Assume $G$ is amenable and let $H \leq G$ be closed. By [F2], $H$ is LCH Hausdorff with its inherited topological-group structure; fix left Haar measures on $G$ and $H$ using [A1, F1]. Hulanicki's criterion [F9] gives $1_{G} ≺ λ_{G}$ . If $Q \subseteq H$ is compact, its image under the continuous inclusion $H ↪ G$ is compact by [F13]. Restricting the coefficient approximations on that compact subset to $H$ shows $1_{H} ≺ λ_{G} ∣_{H}$ : for a scalar $z \in C$ , multiply the approximating vectors for the unit scalar by $z$ to approximate the constant coefficient $∣ z ∣^{2}$ . The restricted-regular-representation lemma [F11] and [F12] show that the restricted unitary representation is strongly continuous and give $λ_{G} ∣_{H} ≺ λ_{H}$ ; transitivity [F10] yields $1_{H} ≺ λ_{H}$ . A second application of [F9] to $H$ proves that $H$ is amenable.

1.3A1F1F3F6F7F8F14construct

Assume $G$ is amenable and $N ⊴ G$ is closed normal. Let $K = G / N$ and $p : G \to K$ . The quotient is LCH Hausdorff by [F3], and has a left Haar measure by [A1, F1]. Let $ν$ be an invariant mean on $L^{\infty} (G)$ and define $m_{K} (ψ) : = ν ([ψ \circ p])$ for $ψ \in U C B (K)$ . Pullback is an actual bounded continuous function. Surjectivity of $p$ gives $∥ ψ \circ p ∥_{\sup} = ∥ ψ ∥_{\sup}$ , and for $g \in G$ , $∥ L_{g} (ψ \circ p) - ψ \circ p ∥_{\sup} = ∥ L_{p (g)} ψ - ψ ∥_{\sup} .$ As $p$ is continuous, the right side tends to zero as $g \to e$ , so pullback maps UCB $(K)$ into UCB $(G)$ . It preserves complex linearity, positivity and the constant one. Thus [F7] and invariance of $ν$ make $m_{K}$ a mean on UCB $(K)$ ; for $k \in K$ choose $g$ with $p (g) = k$ , and the same identity shows $m_{K} (L_{k} ψ) = m_{K} (ψ)$ . By [F8], $K$ satisfies (P1) and is amenable.

1.4A1F1F6F7construct

Assume $N$ and $K = G / N$ are amenable. By [F1] fix left Haar measures, and by [F7] restrict their invariant $L^{\infty}$ means to obtain invariant means $m_{N}$ on UCB $(N)$ and $m_{K}$ on UCB $(K)$ , each with norm one. For $ϕ \in U C B (G)$ and $g \in G$ , define $ϕ_{g} (h) : = ϕ (g h)$ for $h \in N$ . For $n \in N$ and $h \in N$ , $L_{n} ϕ_{g} (h) = ϕ (g n^{- 1} h) = (L_{g n g^{- 1}} ϕ) (g h),$ so $∥ L_{n} ϕ_{g} - ϕ_{g} ∥_{\sup, N} \leq ∥ L_{g n g^{- 1}} ϕ - ϕ ∥_{\sup, G} \to 0$ as $n \to e$ in $N$ . Thus $ϕ_{g} \in U C B (N)$ . Set $F_{ϕ} (g) : = m_{N} (ϕ_{g})$ . For $γ \in G$ and $g \in G$ , $F_{ϕ} (γ^{- 1} g) = m_{N} ((L_{γ} ϕ)_{g})$ ; hence [F7] gives $∥ L_{γ} F_{ϕ} - F_{ϕ} ∥_{\sup, G} \leq ∥ L_{γ} ϕ - ϕ ∥_{\sup, G} \to 0$ as $γ \to e$ . Therefore $F_{ϕ} \in U C B (G)$ .

2.1F3F6F14step 1.4construct

For $n \in N$ , $ϕ_{g n} (h) = ϕ (g n h) = ϕ_{g} (n h) = L_{n^{- 1}} ϕ_{g} (h)$ ; invariance of $m_{N}$ gives $F_{ϕ} (g n) = F_{ϕ} (g)$ . Thus $F_{ϕ}$ is constant on the fibres of $p$ . By [F3] and [F14], it descends to a continuous function $ψ_{ϕ} : K \to C$ with $ψ_{ϕ} \circ p = F_{ϕ}$ . To verify $ψ_{ϕ} \in U C B (K)$ , fix $ε > 0$ . Since $F_{ϕ} \in U C B (G)$ , choose an identity neighbourhood $V$ in $G$ such that $∥ L_{γ} F_{ϕ} - F_{ϕ} ∥_{\sup, G} < ε$ for every $γ \in V$ . The set $p (V)$ is an identity neighbourhood in $K$ by openness of $p$ . For $k \in p (V)$ , take any representative $γ \in V$ with $p (γ) = k$ . Surjectivity of $p$ gives the exact equality $∥ L_{k} ψ_{ϕ} - ψ_{ϕ} ∥_{\sup, K} = ∥ L_{γ} F_{ϕ} - F_{ϕ} ∥_{\sup, G} < ε,$ so $ψ_{ϕ}$ is UCB. The argument uses a representative separately for each estimate and makes no global section choice.

3.1F7F8step 1.4step 2.1algebra∎

Define $M (ϕ) : = m_{K} (ψ_{ϕ})$ for $ϕ \in U C B (G)$ . The construction of $F_{ϕ}$ and descent are complex-linear in $ϕ$ , preserve pointwise nonnegativity, and send $1_{G}$ to $1_{K}$ ; therefore [F7] makes $M$ a positive complex-linear unital mean. For $γ \in G$ , $F_{L_{γ} ϕ} (g) = F_{ϕ} (γ^{- 1} g)$ , so $ψ_{L_{γ} ϕ} = L_{p (γ)} ψ_{ϕ}$ . Invariance of $m_{K}$ now gives $M (L_{γ} ϕ) = M (ϕ)$ . Thus $M$ is a left-invariant UCB mean on $G$ . By [F8], $G$ satisfies Reiter (P1) and is amenable.

Sources

BHV, Kazhdan's Property (T), Appendix G.2, Proposition G.2.2(i)–(ii) and complete proof (printed p. 451) gives quotient and extension inheritance by UCB pullback and fixed points. Appendix G.3, Corollary G.3.4 and Appendix F.1, Proposition F.1.10 with proof (printed pp. 457 and 426) give closed-subgroup inheritance through weak containment of the restricted regular representation. The local proof expands quotient pullback for the library's actual-function UCB and gives an independent UCB-mean averaging proof of the extension clause under its complex-mean convention.

5 · Examples, counterexamples and false statements

None yet.

Amenability Reiter Nets and Folner Conditions

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Complex L∞L∞ space of a locally compact group

Definition

Facts & Assumptions

Proof

Left-invariant means on L∞L∞ of a locally compact group

Definition

Facts & Assumptions

Proof

Amenable locally compact group

Definition

Reiter's condition (P1)

Definition

Facts & Assumptions

Proof

Left Følner nets for locally compact groups

Definition

Facts & Assumptions

Proof

Left-uniformly continuous bounded functions (UCB)

Definition

Facts & Assumptions

Proof

Every locally compact Hausdorff group has an open sigma-compact subgroup

Statement

Facts & Assumptions

Proof

Finite Haar mass, compact detection, and integrable pairings

Statement

Facts & Assumptions

Proof

Remarks

Probability-density averages and locally detectable upper essential values

Statement

Facts & Assumptions

Proof

Remarks

L1 convolution smooths bounded functions into UCB

Statement

Facts & Assumptions

Proof

Sources

A UCB-invariant mean yields a topological invariant mean

Statement

Facts & Assumptions

Proof

Remark

Sources

Probability-density approximation of continuous tests and topological means

Statement

Facts & Assumptions

Proof

Remarks

A topological invariant mean yields norm-approximately invariant densities

Statement

Facts & Assumptions

Proof

Sources

An invariant mean produces a Reiter net

Statement

Facts & Assumptions

Proof

Sources

A Reiter net has an invariant-mean cluster point

Statement

Facts & Assumptions

Proof

Sources

Amenability is equivalent to Reiter's condition (P1)

Statement

Facts & Assumptions

Proof

Sources

Følner nets give Reiter nets

Statement

Facts & Assumptions

Complex $L^{\infty}$ space of a locally compact group

Left-invariant means on $L^{\infty}$ of a locally compact group