Exercise 14, § 1

Let $T$ be a locally compact space, $\mathscr{U}$ a uniform structure on $T$ for which there exists an entourage $V_0$ such that $V_0(t)$ is compact for all $t \in T$ (GT, II, §4, Exer. 9). On the other hand, let $\Gamma$ be a uniformly equicontinuous group of homeomorphisms of $T$ (for the uniform structure $\mathscr{U}$), such that there exists an $a \in T$ whose orbit under $\Gamma$ is dense. Show that there exists on $T$ a positive measure $\neq 0$ invariant under $\Gamma$, and that this measure is unique up to a constant factor. One may proceed as follows:

$1^\circ$ As for the existence of the invariant measure, follow the method of No. 2, Th. 1; one proves first that if $K$ is a compact subset of $T$ and $U$ is an open neighborhood of $a$, there exists a finite number of elements $\sigma_i \in \Gamma$ such that $K \subset \bigcup_i \sigma_i(U)$.

$2^\circ$ As for uniqueness, observe first that the entourages of $\mathscr{U}$ that are invariant under every homeomorphism $\sigma \times \sigma$ of $T \times T$, for $\sigma \in \Gamma$, form a fundamental system of entourages $\mathscr{S}$. For every relatively compact set $A \subset T$ and every relatively compact set $B \subset T$ with nonempty interior, let $(A : B)$ be the smallest number of elements of a covering of $A$ by sets of the form $\sigma B$, where $\sigma \in \Gamma$; if $C \subset T$ is a third relatively compact set with nonempty interior, then $(A : C) \leq (A : B)(B : C)$. Let $K$ be a compact subset of $T$, $L \supset K$ a relatively compact open set, $V \in \mathscr{S}$ a symmetric open entourage contained in $V_0$ such that $V(K) \subset L$, and $W \in \mathscr{S}$ a symmetric closed entourage contained in $V$. For every symmetric entourage $U \in \mathscr{S}$ such that $UW \subset V$ and $WU \subset V$, show that, for every positive measure $\nu$ on $T$ invariant under $\Gamma$,

(1)

(2)

For this, one proves that if $(\sigma_i(U(a)))$ is a covering of $L$ formed by $(L : U(a))$ sets, every $x \in K$ belongs to at least $(W(a) : U(a))$ sets of the form $\sigma_i(V(a))$, and that every $y \in L$ belongs to at most $(V(a) : U(a))$ sets of the form $\sigma_i(W(a))$ (note that if $y \in \sigma_i(W(a))$, then $\sigma_i(U(a))$ is contained in $V(x)$).

Let $\mathscr{F}$ be an ultrafilter finer than the section filter of $\mathscr{S}$; let $A_0$ be a nonempty, relatively compact open set in $T$, and set $\lambda_U(A) = (A : U(a))/(A_0 : U(a))$ for every symmetric entourage $U \in \mathscr{S}$ and every relatively compact subset $A$ of $T$. Let $\lambda(A) = \lim_{\mathscr{F}} \lambda_U(A)$; for every compact subset $K$ of $T$, set $\lambda'(K) = \inf \lambda(B)$, where $B$ runs over the set of relatively compact open neighborhoods of $K$. Deduce from (1) and (2) that

Conclude from this that if $K_1, K_2$ are two compact subsets of $T$, and $L_1 \supset K_1$, $L_2 \supset K_2$ are two relatively compact open sets, then

and finally that $\lambda'(K_1)\nu(K_2) = \lambda'(K_2)\nu(K_1)$.