Exercise 12, § 2

a) Let $X$ be a topological space, $f : X \to \mathbf{N}$ an upper semi-continuous function. Let $X_0$ be the set of points of $X$ where $f$ is locally constant, and $Y_0 = X - X_0$. Define recursively two sequences $(X_i)_{i \geq 0}$, $(Y_i)_{i \geq 0}$ of subsets of $X$, in the following way: $X_i$ is the set of points of $Y_{i-1}$ where $f|Y_{i-1}$ is locally constant, and $Y_i = Y_{i-1} - X_i$. Show that $X_i$ is a dense open subset of $Y_{i-1}$, and that $\bigcap_i Y_i = \varnothing$.

(Show that $f(x) > i$ at every point of $Y_i$.)

b) Let $X$ be a set, $H$ a group operating on the right in $X$, $\pi$ the canonical mapping of $X$ onto $X/H$, and $\mathscr{A}$ (resp. $\mathscr{B}$) the set of $A \subset X$ such that $\pi|A : A \to X/H$ is injective (resp. surjective). Let $(V_1, V_2, \ldots)$ be a countable covering of $X$ by elements of $\mathscr{A}$. Let

Show that $F = \bigcup_{i \geq 0} V'_i \in \mathscr{A} \cap \mathscr{B}$.

c) Let $X$ be a locally compact space, $H$ a countable discrete group operating on the right in $X$ continuously and properly, $\pi$ the canonical mapping of $X$ onto $X/H$, and $\mu$ a measure $\geq 0$ on $X$. Show that if the sets $V_i$ of b) are quadrable (Ch. IV, §5, Exer. 17 d)), then the set $F$ of b) is a quadrable fundamental domain.

d) Let us maintain the hypotheses of c), and use the notations $H_x, n(x)$ of No. 10. An $x \in X$ is said to be general if there exists a neighborhood $V$ of $x$ in $X$ such that $H_y = H_x$ for all $y \in V$. Show that the general points of $X$ are those where the function $n$ is locally constant. Show that a general point admits an open neighborhood that belongs to $\mathscr{A}$ and is quadrable (make use of Ch. IV, §5, Exer. 17 d)).

e) Let us maintain the hypotheses of c), and assume moreover that $X$ is countable at infinity. Show that if $X - X_0$ is $\mu$-negligible, then there exists a fundamental domain $F$ that is Borel and quadrable. (Apply the construction of a) to the function $n$. Then apply the results of c) and d) in $X_0$. Argue similarly in each $X_i$.)

f) Let $U \in \mathscr{B}$ be an open set. Show that one can impose on the set $F$ of e) the following supplementary properties: 1) $F \subset U$; 2) for every compact subset $K$ of $X$, the set of $s \in H$ such that $Fs$ intersects $K$ is finite.