Definition 1

Let $X$ be a locally compact space, $\mu$ a measure on $X$. A mapping $f$ of $X$ into a topological space $F$ is said to be measurable for the measure $\mu$ (or to be $\mu$-measurable) if, for every compact subset $K$ of $X$, there exist a $\mu$-negligible set $N \subset K$ and a partition of $K - N$ formed by a sequence (finite or infinite) $(K_n)$ of compact sets, such that the restriction of $f$ to each $K_n$ is continuous.

It is clear that every continuous mapping of $X$ into $F$ is measurable.

Note that if $\mu$ and $\nu$ are two measures on $X$ such that every $\mu$-negligible set is $\nu$-negligible, then every $\mu$-measurable function is also $\nu$-measurable (cf. Ch. V, §5, Nos. 5, 6).

Definition 1 may be transformed into the following criterion: