Example

— Let $L$ be a compact subset of $T$, $\lambda$ a measure on $L$, and $\mu$ the measure on $T$ defined by $\lambda$ (No. 3, Example 2). A function $f$ defined on $T$ is $\mu$-measurable if and only if $f_L$ is $\lambda$-measurable. For, this condition is obviously necessary. Conversely, if it is satisfied, there exists a partition of $L$ into a $\lambda$-negligible set $N$ and a sequence $(L_n)$ of compact sets, such that $f_{L_n}$ is continuous for all $n$. If $K$ is a compact subset of $T$, the set $K - \bigcup^n (K \cap L_n)$ has an intersection with $L$ that is $\lambda$-negligible, hence this set is $\mu$-negligible by formula (3) of No. 3, and the restriction of $f$ to $K \cap L_n$ is continuous for all $n$.

Def. 7 permits extending, without a new proof, a number of results on measurable functions to the case of spaces not locally compact. Here are some of them, which we shall use henceforth without further reference: the open sets and the closed sets of $T$ are $\mu$-measurable; the $\mu$-measurable sets form a tribe (Ch. IV, §5, No. 4, Cor. 2 of Th. 2), that contains the Borel sets of T (loc. cit., Cor. 3), and the Souslin sets (Ch. IV, §5, No. 1, Cor. 2 of Prop. 3)(1). The usual algebraic operations on numerical functions preserve measurability (Ch. IV, §5, No. 3), as do the operations of countable passage to the limit (loc. cit., No. 4, Th. 2 and Cor. 1). The following property merits more explicit mention: