Theorem 1

Let X be a locally compact space, $\mu$ a measure on X, $(F_n)$ a sequence of topological spaces, $F = \prod_n F_n$ their product. For each index n, let $f_n$ be a measurable mapping of X into $F_n$, and let $f(x) = (f_n(x)) \in F$; then, for every continuous mapping u of $f(X)$ into a topological space G, the function $x \mapsto u(f(x))$ is measurable.

For every compact subset K of X and every number $\varepsilon > 0$, there exists a compact set $K_1 \subset K$ such that $|\mu|(K - K_1) \leq \varepsilon$ and such that the restrictions to $K_1$ of all the functions $f_n$ are continuous (No. 1, Prop. 2); it is clear that $u \circ f$ is continuous on $K_1$, whence the theorem.