Exercise 19, § 8

Let $X$ be a locally compact space, $Y$ a locally compact space countable at infinity, $A$ a universally measurable subset of $X \times Y$.

a) For every $x \in X$, show that the section $A(x)$ is a universally measurable subset of $Y$. Moreover, for every positive measure $\mu$ on $Y$, the function $x \mapsto \mu^*(A(x))$ is universally measurable on $X$ (make use of the Lebesgue–Fubini theorem).

b) Let $\mu$ be a positive measure on $Y$ such that, for almost every $y \in Y$ (for $\mu$), the section $A(y)$ of $A$ is countable. Show that the set $N$ of $x \in X$ such that $\mu^*(A(x)) > 0$ cannot contain any countable compact set.