Corollary 2

Let $X$ be a locally compact space that is countable at infinity. If $f$ is a measurable mapping of $X$ into a metrizable space $F$, then there exists a sequence $(g_n)$ of measurable step functions, with values in $F$, such that $g_n(x)$ tends to $f(x)$ for almost every $x \in X$.

If $X$ is the union of an increasing sequence $(A_n)$ of compact sets, the sets $A_n - A_{n-1}$ that are nonempty form a partition of $X$ into integrable sets; consequently, there exist a negligible set $N \subset X$ and a partition of $\mathbf{C}N$ formed by a sequence $(K_n)$ of compact sets such that the restriction of $f$ to each $K_n$ is continuous; the proof can then be concluded as in Th. 3 without modification.