Remark

— 1) If $X$ is a Souslin space, then every compact subspace of $X$ is a Souslin space, hence is metrizable (TG, IX, Appendix I, Cor. 2 of Prop. 3),$^{(2)}$

(1) In case $f \cdot \varphi_A$ is not universally measurable, make use of Remark 2) below. (2) This result does not appear in GT (cf. the footnote to Remark 1 of §1, No. 9).

and every measure on X is moderated (§ 1, No. 9, Remark 1). By Prop. 13, every measure $\nu$ on X therefore admits a disintegration with respect to every $\nu$-proper mapping.

2) With the notations of Prop. 13, let $f$ be a positive $\nu$-measurable function. One can prove, as in Ch. V, § 3, No. 2, Prop. 5, that the set of $t \in T$ such that $f$ is not $\lambda_t$-measurable is locally $\mu$-negligible, that $t \mapsto \lambda_t^\bullet(f)$ is $\mu$-measurable, and that the relation (12) again holds.