Exercise 26, § 2

Let S, T be two compact spaces, f a finite numerical function defined on S × T.

a) In order that the mapping s ↦ f(s, ·) of S into $\mathbf{R}^T$ be a continuous mapping of S into the space $\mathscr{C}(T)$ equipped with the topology $\sigma(\mathscr{C}(T), \mathscr{M}(T))$, it is necessary and sufficient that f be bounded and that each of the partial mappings $f(s, \cdot), f(\cdot, t)$ ($s \in S, t \in T$) be continuous. (To see that the condition is sufficient, first show that the image M of S under $s \mapsto f(s, \cdot)$ is compact for $\sigma(\mathscr{C}(T), \mathscr{M}(T))$. For this, observe that $s \mapsto f(s, \cdot)$ is continuous when $\mathscr{C}(T)$ is equipped with the topology of pointwise convergence on T; use Eberlein’s theorem (TVS, IV, §5, No. 3, Th. 1) to reduce to proving that every sequence $(f(s_n, \cdot))$ has a cluster point in $\mathscr{C}(T)$ for $\sigma(\mathscr{C}(T), \mathscr{M}(T))$. Reduce to the case that T is metrizable, by considering a suitable quotient space of T (cf. Exer. 10 a)), and observe that, on a subset of $\mathscr{C}(T)$ relatively compact for the topology of pointwise convergence on T, this topology is identical to the topology of pointwise convergence on a dense subset of T. Obtain thus a subsequence of $(f(s_n, \cdot))$ that converges pointwise on T, and apply Lebesgue’s theorem. Finally, observe that on M, the topology $\sigma(\mathscr{C}(T), \mathscr{M}(T))$ and the topology of pointwise convergence are identical.)

b) Assume that each of the partial mappings $f(s, \cdot), f(\cdot, t)$ is continuous ($s \in S, t \in T$). Show that for every positive measure $\mu$ on S and every $\varepsilon > 0$, there exists a compact subset $K \subset S$ such that $\mu(S - K) \leq \varepsilon$ and the restriction of f to $K \times T$ is continuous. (Reduce to the case that f is bounded; apply a) and Exer. 25 c).)

c) With the same hypotheses as in b), show that f is measurable for every measure $\nu$ on $S \times T$. (Apply b) to the image of $\nu$ under the projection of $S \times T$ onto S.)