Proposition 2

Let $(F_n)$ be a sequence of topological spaces and, for each $n$, let $f_n$ be a measurable mapping of $X$ into $F_n$. For every compact set $K \subset X$ and every $\varepsilon > 0$, there exists a compact set $K_0 \subset K$ such that $|\mu|(K - K_0) \leq \varepsilon$ and the restriction to $K_0$ of each of the functions $f_n$ is continuous.

For each integer $n \geq 1$ there exists a compact set $K_n \subset K$ such that $|\mu|(K - K_n) \leq \varepsilon / 2^n$ and the restriction of $f_n$ to $K_n$ is continuous. The set $K_0 = \bigcap_{n=1}^\infty K_n$ is compact, the restrictions to $K_0$ of all of the functions $f_n$ are continuous and, since $K - K_0$ is contained in the union of the $K - K_n$, $|\mu|(K - K_0) \leq \sum_{n=1}^\infty \varepsilon / 2^n = \varepsilon$.