Corollary 1

Let $f$ be a measurable mapping of $X$ into a Banach space $F$; for every compact set $K \subset X$, there exists a sequence $(g_n)$ of measurable step functions, with supports contained in $K$, such that $|g_n(x)| \leq |f(x)|$ for all $x \in X$ and such that $g_n(x)$ tends to $f(x)$ for almost every $x \in K$.

With notations as in the proof of Th. 3, and denoting by $a_{ij}$ one of the values of $f$ in $A_{ij}$, it suffices to take, as the value of $g_n$ in $A_{ij}$, the point 0 if $|a_{ij}| \leq 1/n$ and the point $a_{ij}(1 - 1/(n|a_{ij}|))$ otherwise; finally, set $g_n(x) = 0$ on the complement of the union of the $A_{ij}$ ($1 \leq i \leq n,\ 1 \leq j \leq q_i$).