Exercise 5, § 8

Let T, T' be two locally compact spaces, $\mu$ a positive measure on T, $\mu'$ a positive measure on T'; suppose that every compact subset of T is metrizable. Let $f$ be a mapping of $T \times T'$ into a metrizable space G, such that: 1° for every $t \in T$, the mapping $t' \mapsto f(t, t')$ is $\mu'$-measurable; 2° for every $t' \in T'$, the mapping $t \mapsto f(t, t')$ is continuous. Show that, under these conditions, $f$ is $(\mu \otimes \mu')$-measurable. (Observe (using Egoroff's theorem) that for every compact subset K of T, the restriction of $f$ to $K \times T'$ is the limit of a sequence of $(\mu \otimes \mu')$-measurable functions.)