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A transitive Borel $G$ -space with a quasi-invariant measure class is ergodic

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group, $H \leq G$ a closed subgroup, and $μ$ a nonzero quasi-invariant Radon measure on $G / H$ . If $E \subseteq G / H$ is Borel with $μ (E △ g E) = 0$ for every $g \in G$ , then $μ (E) = 0$ or $μ ((G / H) ∖ E) = 0$ . Equivalently, any transitive system of imprimitivity on $G / H$ whose measure class is the quasi-invariant class is ergodic.

Facts & Assumptions

Given: AC, the group $G$ , closed subgroup $H$ , the quotient $q : G \to G / H$ , a nonzero quasi-invariant Radon measure $μ$ on $G / H$ , and a Borel $E \subseteq G / H$ with $μ (E △ g E) = 0$ for all $g$ .

[F1]

$q$ is continuous and open, $G / H$ is a standard Borel $G$ -space with Borel action, $q^{- 1} (g E) = g q^{- 1} (E)$ for the left action, and a Borel set $F \subseteq G / H$ is $μ$ -null if and only if $q^{- 1} (F)$ is Haar null (Compact lifts and averaging onto C_c(G/H), Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Haar null classes and Borel descent on a homogeneous space, Left group actions, transitive actions, and faithful actions).

[F2]

There exists a full-support strongly quasi-invariant rho-measure $μ_{ρ}$ whose class is the quasi-invariant class, and every nonzero quasi-invariant $σ$ -finite Borel measure is equivalent to $μ_{ρ}$ ; Radon measures on the $σ$ -compact space $G / H$ are $σ$ -finite (Existence of rho-functions and quotient measure classes, Quasi-invariant Radon measure on G/H, Haar null classes and Borel descent on a homogeneous space, Radon measure on an LCH space, Countably compact, Lindel"of, sequentially compact, limit point compact and $σ$ -compact spaces, and relatively compact subsets, Second countability: an at most countable basis for the topology, 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]

Tonelli applies to nonnegative product-measurable functions on sigma-finite measure spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product). Right translation by $t$ scales left Haar measure by a positive scalar, so it preserves Haar-null Borel sets; this holds at the Borel-measure level, not only for $C_{c}$ integrals (Modular function of a locally compact group, Borel-level form). Left Haar measure on $G$ is sigma-finite because $G$ is sigma-compact by [F1].

[F4]

For a transitive system of imprimitivity on $G / H$ , invariance of a spectral projection $P (B)$ under the representation means $P (g B) = P (B)$ for all $g$ , and $P$ is strongly countably additive, so $P (B △ g B) = 0$ whenever $P (g B) = P (B)$ (Systems of imprimitivity for a Borel $G$ -space, Scalar and complex measures from a pvm).

[F5]

AC is the standing hypothesis (The Axiom of Choice, Topological group: multiplication and inversion are continuous).

Proof

technique · direct

Given: AC, the data and the invariant Borel set $E$ of the statement.

1.1F1F2

Lift the indicator: $f (t) : = 1_{E} (q (t))$ . For every $g$ , applying the hypothesis to $g^{- 1}$ gives $μ (E △ g^{- 1} E) = 0$ , so by [F1] the set $q^{- 1} (E △ g^{- 1} E)$ is Haar null. Since $q (g t) = g q (t)$ , one has $f (g t) = 1_{E} (g q (t)) = 1_{g^{- 1} E} (q (t))$ , which equals $f (t) = 1_{E} (q (t))$ at every $t$ outside $q^{- 1} (E △ g^{- 1} E)$ ; hence $f \circ L_{g} = f$ Haar-a.e. for every $g \in G$ .

2.1F1F3step 1.1

The function $(g, t) \mapsto ∣ f (g t) - f (t) ∣$ is Borel and hence product-measurable, since multiplication is continuous and the Borel sigma-algebra of a product of second-countable spaces is the product Borel sigma-algebra. By step 1.1 each integral in $t$ is zero. Tonelli therefore gives $\int_{G} \int_{G} ∣ f (g t) - f (t) ∣ d g d t = 0$ , so for Haar-almost every $t$ , $f (g t) = f (t)$ for Haar-almost every $g$ .

3.1F3step 2.1

Choose $t_{0}$ with the preceding property; the conull set is nonempty since Haar measure is nonzero. The exceptional set of $g$ is Haar null, and its right translate by $t_{0}$ remains null by [F3]. Substituting $u = g t_{0}$ therefore gives $f (u) = f (t_{0})$ for Haar-almost every $u$ . Since $f (t_{0}) \in {0, 1}$ , $f = 1_{q^{- 1} (E)}$ is Haar-a.e. zero or Haar-a.e. one.

4.1F1step 3.1

By the null-class equivalence of [F1], $f = 0$ Haar-a.e. gives $μ (E) = 0$ and $f = 1$ Haar-a.e. gives $μ ((G / H) ∖ E) = 0$ . This proves the first assertion.

5.1F4step 4.1F5∎

Equivalence with ergodicity of transitive systems: let $(U, P)$ be a transitive system on $G / H$ whose null class (the class of $P$ -null Borel sets) is the quasi-invariant class, and let $P (B)$ be invariant under $U$ . Then $P (g B) = P (B)$ for all $g$ , so $P (B △ g B) = 0$ by [F4] and hence $B △ g B$ is null for every measure in the quasi-invariant class; applying [step 4.1] to a representative $μ$ gives $μ (B) = 0$ or $μ (B^{c}) = 0$ , and translating back gives $P (B) = 0$ or $P (B) = I$ . Thus the system is ergodic.

Depends on

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90 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. W. Mackey, Imprimitivity for Representations of Locally Compact Groups I, PNAS 35 (1949) 537-545 (Internet Archive capture of the PubMed Central scan) (standard reference, not scraped)
G. B. Folland, A Course in Abstract Harmonic Analysis, Chapter 2 §2.6 (transitive quasi-invariant actions are ergodic) (standard reference, not scraped)

A transitive Borel GG-space with a quasi-invariant measure class is ergodic