Corollary

— Let $(A_n)$ be a sequence of pairwise disjoint measurable subsets, with union A. For every subset B of T,

and in particular

Among the properties of measurable functions or sets that extend as above to Hausdorff spaces, we cite also Prop. 12 of Ch. IV, §5, No. 8 ($\mu$-dense families of compact sets). Thus, a function $f$ with values in a topological

(1) The proof of this corollary is valid without modification for Souslin sets in a nonmetrizable locally compact space (GT, IX, §6, No. 9, Th. 5).

space (Hausdorff or not) is $\mu$-measurable if and only if the set of compact subsets K of T, such that $f_K$ is continuous, is $\mu$-dense (loc. cit., No. 10, Prop. 15).