LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Haar candidate sets have the finite intersection property

Statement

Assume AC and fix $0 \neq f_{0} \in C_{c} (G)^{+}$ . Put $P = \prod_{0 \neq f \in C_{c} (G)^{+}} [1 / (f_{0} : f), (f : f_{0})]$ . For each open identity neighbourhood $U$ , let $E_{U}$ be the closure in $P$ of the vectors $I_{ϕ}$ with $0 \neq ϕ \geq 0$ and $supp ϕ \subseteq U$ . These closed sets are nonempty and have the finite intersection property, and $⋂_{U} E_{U} \neq \emptyset$ . Every common point, extended by value zero at $0$ , is additive, positively homogeneous, strictly positive on nonzero nonnegative functions, normalized at $f_{0}$ , and left invariant.

Facts & Assumptions

Given: AC, $f_{0}$ , the product and closed candidate sets as stated.

[F1]

Approximant vectors obey closed coordinate bounds and the exact normalization, homogeneity and invariance equations. (Normalized approximate Haar functionals are positive and invariant in the limit)

[F2]

All sufficiently small-support approximants have arbitrarily small additivity error. (Haar covering functionals are asymptotically additive)

[F3]

Under AC a product of compact spaces is compact. (Tychonoff's theorem: an arbitrary product of compact spaces is compact in the product topology, assuming the Axiom of Choice)

[F4]

Closed sets with the FIP in a compact space have a common point. (A space is compact exactly when every family of closed subsets with the finite intersection property has nonempty intersection)

[F5]

Cutoffs at a compact singleton exist under DC. (LCH Urysohn cutoff)

[F6]

AC is assumed for the compact product and inherited cutoffs. (The Axiom of Choice)

Proof

technique · direct

1.1F1F5F6

For any $U$ , a cutoff $v$ at ${e} \subseteq U$ has $v (e) = 1$ , $0 \leq v \leq 1_{U}$ and compact support. Set $ϕ = (v - 1 / 2)^{+}$ . Then $ϕ (e) = 1 / 2$ and its support lies in the compact set ${v \geq 1 / 2} \subseteq U$ . Thus $E_{U}$ contains an approximant. The coordinate intervals are nonempty because they contain every such vector, and they are finite closed real intervals.

2.1F3F4F6step 1.1

For finitely many neighbourhoods $U_{1}, \dots, U_{n}$ , their intersection $W$ is an identity neighbourhood, and $E_{W} \subseteq ⋂_{j} E_{U_{j}}$ . Step 1.1 makes this intersection nonempty. For $n = 0$ the intersection is $P$ , also nonempty by an approximant. The compact real intervals and AC give compactness of $P$ , so the closed FIP theorem supplies a common point $T$ .

3.1F1F2step 2.1∎

The closed equations in [F1] hold throughout each $E_{U}$ , hence for $T$ , and its strictly positive coordinate lower bounds persist. Set $T (0) = 0$ . For $f, g \geq 0$ and $ϵ > 0$ , [F2] gives $U$ such that $0 \leq I_{ϕ} (f) + I_{ϕ} (g) - I_{ϕ} (f + g) < ϵ$ on its approximants. This finite-coordinate condition with upper bound $\leq ϵ$ is closed, so it holds for $T \in E_{U}$ . An error in $[0, ϵ]$ for every positive $ϵ$ is zero. Thus $T$ is additive; when an argument is zero this is already the assigned zero value.

Sources

Knapp, Advanced Real Analysis, VI §2, pp.225–230, Lemmas 6.9–6.13. Local argument and conventions as displayed above.

Depends on

Dependency tree · 2 levels

Sources

Knapp, Advanced Real Analysis, VI §2, pp.225–230, Lemmas 6.9–6.13