Definition 2

A mapping f of T into a topological space F is said to be universally measurable if it is $\mu$-measurable for every positive measure $\mu$ on T.

The subsets of T whose characteristic function is universally measurable are called universally measurable sets. They form a tribe on T (Ch. IV, §5, No. 4, Cor. 2 of Th. 2) that contains the Borel sets (same ref., Cor. 3), and the Souslin sets if T is metrizable (Ch. IV, §5, No. 1, Cor. 2 of Prop. 3). For a mapping $f$ of T into a topological space F, metrizable and separable, to be universally measurable, it is necessary and sufficient that the inverse image under $f$ of every closed ball in F be a universally measurable subset of T (Ch. IV, §5, No. 5, Th. 4).