Exercise 27, § 1

Let G be a group operating on a set X. A subset P (resp. C) of X is said to be a G-filling (resp. G-covering) if, for every $s \neq e$ in G, one has $sP \cap P = \varnothing$ (resp. if $X = \bigcup_{s \in G} sC$). One calls G-paving a subset P that is both a G-filling and a G-covering.

a) Assume that X is locally compact, that G is countable, operates continuously in X, and that there exists a nonzero positive measure $\mu$ on X invariant under G. Let P and C be a G-filling and a G-covering that are $\mu$-integrable. Show that $\mu(C) \geq \mu(P)$. (Observe that $\mu(C) \geq \sum_{s \in G} \mu(C \cap sP)$.)

b) Assume in addition that there exists on X a uniform structure compatible with the topology of X and admitting a fundamental system $\mathscr{G}$ of open entourages invariant under G. On the other hand, one denotes by $\Delta(G)$ the infimum of the numbers $\mu(C)$ over all the integrable G-coverings C of X. Let V be an entourage belonging to $\mathscr{G}$, and let $a \in X$ be such that $\mu(V(a)) > \Delta(G)$; show that there exists an $s \in G$ such that $s \neq e$ and $(a, sa) \in V^{-1} \circ V$.

c) Assume that X is a locally compact group, $\mu$ a left Haar measure, and G a countable subgroup of X, operating by left translation. With the same definition of $\Delta(G)$ as in b), show that if A is an integrable subset of X such that $\mu(A) > \Delta(G)$, then there exists $s \in G \cap AA^{-1}$ such that $s \neq e$. Special case that $X = \mathbf{R}^n$ and G is a discrete subgroup of rank n in $\mathbf{R}^n$: $\Delta(G)$ is then equal to the absolute value of the determinant of a basis of G over $\mathbf{Z}$ with respect to the canonical basis of $\mathbf{R}^n$; if A is a symmetric closed convex set with nonempty interior$^5$ in $\mathbf{R}^n$ such that $\mu(A) \geq 2^n \Delta(G)$, show that there exists a point of $A \cap G$ distinct from 0 (Minkowski's theorem).

d) With hypotheses as in a), let f be a function $\geq 0$ and $\mu$-integrable on X. Show that there exist two points $a, b$ of X such that $\mu(C) \sum_{s \in G} f(sa) \geq \int_X f(x) d\mu(x)$ and $\mu(P) \sum_{s \in G} f(sb) \leq \int_X f(x) d\mu(x)$. (Observe that if g is an integrable function $\geq 0$, E a $\mu$-integrable set in X, then there exists a $c \in E$ such that $\int_E g(x) d\mu(x) \leq g(c)\mu(E)$, and a $c' \in E$ such that $\int_E g(x) d\mu(x) \geq g(c')\mu(E)$.)