Proposition 7

Let $f$ be a measurable mapping of $X$ into a topological space $F$; the inverse image under $f$ of every closed (resp. open) set in $F$ is measurable.

It suffices to carry out the proof for the inverse image $^{-1}f(A)$ of a closed set $A$ in $F$. Let $K$ be a compact subset of $X$; there exist a negligible set $N \subset K$ and a partition $(K_n)$ of $K - N$ formed of compact sets such that the restriction of $f$ to each $K_n$ is continuous. The intersection $K_n \cap ^{-1}f(A)$ is thus the inverse image of the closed set $A$ under the restriction of $f$ to $K_n$; it is therefore a closed set in $K_n$, hence is compact. The set $K \cap ^{-1}f(A)$ is therefore the union of the negligible set $N \cap ^{-1}f(A)$ and the compact sets $K_n \cap ^{-1}f(A)$, which proves that $^{-1}f(A)$ is measurable.