Exercise 6, § 8

a) Let T be a locally compact space, $\nu$ a positive measure on T, I an interval of $\mathbf{R}$, $\mu$ a positive measure on I. Let A be a subset of the product $I \times T$, such that: 1° for every $x \in I$, the section $A(x) \subset T$ of A at x is $\nu$-measurable; 2° the relation $x \leq y$ implies $A(x) \subset A(y)$. Show that A is $(\mu \otimes \nu)$-measurable. (Reduce to the case that I and T are compact and $\mu$ is diffuse; consider an increasing sequence $(x_i)_{0 \leq i \leq n}$ of points of I, $x_0$ and $x_n$ being the end-points of I, such that $\mu([x_i, x_{i+1}]) \leq \varepsilon$ for $0 \leq i \leq n-1$, and form two subsets B, C of $I \times T$, measurable for $\lambda = \mu \otimes \nu$ and such that $B \subset A \subset C$ and $\lambda(C - B) \leq \varepsilon \nu(t)$.)

b) Deduce from a) that if $f$ is a numerical function defined on $I \times T$, such that: 1° for each $x \in I$, $t \mapsto f(x, t)$ is $\nu$-measurable, and 2° for each $t \in T$, $x \mapsto f(x, t)$ is increasing, then $f$ is $(\mu \otimes \nu)$-measurable.

c) Let $g$ be a numerical function defined on $I \times T$, such that: 1° for each $x \in I$, $t \mapsto g(x, t)$ is $\nu$-measurable; 2° for each $t \in T$, $x \mapsto g(x, t)$ is monotone. Deduce from b) that $g$ is $(\mu \otimes \nu)$-measurable. (Let $T_1 \subset T$ be the set of $t \in T$ such that the mapping $x \mapsto g(x, t)$ is increasing; show that $T_1$ is $\nu$-measurable. To that end, for every pair $(r_1, r_2)$ of rational numbers belonging to I such that $r_1 \leq r_2$, consider the set $D_{r_1, r_2}$ of points $t \in T$ such that $g(r_1, t) \leq g(r_2, t)$ and express $T_1$ as the intersection of sets $D_{r_1, r_2}$.)