Definition 7

A function $f$ defined on $T$, with values in a topological space $F$ (Hausdorff or not) is said to be measurable for the measure $\mu$ (or to be $\mu$*-measurable)* if, for every compact subset $K$ of $T$, the function $f_K$ is $\mu_K$*-measurable*.

This amounts to saying that there exists, for every compact set $K$, a partition of $K$ into a $\mu_K$-negligible set $N$ and a sequence $(K_n)$ of compact sets, such that the restriction of $f$ to each $K_n$ is continuous. Since it is equivalent to say that $N$ is $\mu_K$-negligible or that it is locally $\mu$-negligible (No. 4), one sees that $f$ is $\mu$-measurable if and only if, for every compact set $K$, there exists a partition of $K$ into a locally $\mu$-negligible set $N$ and a sequence $(K_n)$ of compact sets such that $f_{K_n}$ is continuous for all $n$. This definition is identical to that of Def. 1 of Ch. IV, §5, No. 1, and one thus recovers the usual concept of measurable function when $T$ is locally compact.

A subset $A$ of $T$ is said to be measurable if its characteristic function is measurable. When $A$ is $\mu$-measurable and $\mu^\bullet(A) < +\infty$, this number is denoted simply $\mu(A)$ and is called the measure of $A$. One similarly writes $\mu(f)$ for $\mu^\bullet(f)$ when $f$ is $\geqslant 0$, $\mu$-measurable, and $\mu^\bullet(f) < +\infty$.

If $\theta$ is a complex measure on $T$, a function $f$ (resp. a subset of $T$) is said to be $\theta$-measurable if it is measurable for the positive measure $|\theta|$. The results below may be extended to complex measures.