Exercise 26, § 2

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

(a) For the mapping $s \to f(s, .)$ from S into $\mathbf{R}^T$ to be a continuous mapping from S into the space $C(T)$ endowed with the topology $\sigma(C(T), \mathscr{M}(T))$, it is necessary and sufficient that f be bounded and that each of the partial mappings $f(s, .), f(., t)$ ($s \in S, t \in T$) be continuous. (To see that the condition is sufficient, first prove that the image M of S by $s \rightarrow f(s, .)$ is compact for $\sigma(C(T), \mathscr{M}(T))$. For this, remark that $s \rightarrow f(s, .)$ is continuous when $C(T)$ is endowed with the topology of simple convergence in T; use Eberlein’s th. (Esp. vect. top., chap. IV, § 2, exerc. 15) to reduce oneself to proving that every sequence $(f(s_n, .))$ has a value of adherence in $C(T)$ for $\sigma(C(T), \mathscr{M}(T))$. Reduce oneself to the case where T is metrizable, by considering a suitable quotient space of T (cf. exerc. 10 a)), and remark that, on a part of $C(T)$ relatively compact for the topology of simple convergence in T, this topology is identical with the topology of simple convergence in a everywhere dense part of T. Obtain in this way an extracted sequence from $(f(s_n, .))$ which converges simply in T, and apply Lebesgue’s th. Finally, remark that on M the topology $\sigma(C(T), \mathscr{M}(T))$ and the topology of simple convergence are identical.)

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

(c) The hypotheses being the same as in b), show that f is measurable for every measure ν on S × T. (Apply b) to the image of ν by the projection of S × T onto S.)