Exercise 10, § 3

Let K be a metrizable compact space, $\mu$ a positive measure on K, and $f$ a $\mu$-measurable mapping of K into a Hausdorff topological space E. Denote by I (resp. U) the subset of E formed by the $y$ such that $f^{-1}(y)$ is infinite (resp. uncountable). a) Show that there exists in K a $\mu$-measurable set H such that $f(H) = I$ and such that, for every $y \in E$, $f^{-1}(y) \cap CH$ is finite. (Let $(K_n)$ be an increasing sequence of compact subsets of K such that the restriction of $f$ to each $K_n$ is continuous and the complement N of the union F of the $K_n$ is $\mu$-negligible; for every $k > 1$, let $B_k$ be the subset of E formed by the $y$ such that $F \cap f^{-1}(y)$ contains at least $k$ distinct points, and let $A_k = F \cap f^{-1}(B_k)$; take for H the union of the set $A = \bigcap_k A_k$ and the set $N \cap f^{-1}(I)$, and use Exer. 9.) b) Let $\mathfrak{F}$ be the set of $\mu$-measurable sets $M \subset H$ such that $M \cap f^{-1}(y)$ is finite for every $y \in E$; let $\alpha$ be the supremum of the measures $\mu(M)$ for $M \in \mathfrak{F}$; there exists an increasing sequence $(M_n)$ of sets in $\mathscr{F}$ such that $\lim_{n \to \infty} \mu(M_n) = \alpha$. Let $P = \bigcup_n M_n$, and let $S$ be a measurable section of $H \cap \mathbf{C}P$ for the equivalence relation $f(x) = f(y)$ (Th. 3); show that $\mu(S) = 0$.

c) Show that there exists a $\mu$-negligible set $Z \subset K$ such that $f(Z) = U$ and that, for every $\varepsilon > 0$, there exists a $\mu$-measurable set $L \subset K$ such that $\mu(L) \leq \varepsilon$ and $f(L) = I$ (observe that $f(H \cap \mathbf{C}M_n) = I$ for every $n$ and that $U \subset f(S)$).