Proposition 6

For a mapping $f$ of T into a topological space F to be universally measurable, it is necessary and sufficient that $f$ be measurable for every positive measure on T with compact support.

The condition is obviously necessary; on the other hand it is sufficient, because every positive measure $\mu$ is the sum of a family of measures with compact support (§2, No. 3, Prop. 4): the statement then follows from Prop. 2 of §2, No. 2.