Proposition 12

— Let $f$ be a $\nu$-measurable positive function defined on $X = S \times T$. Assume that $f$ is $\nu$-moderated (resp. that $\mu$ is moderated). Then: a) The set $N$ of $s \in S$ such that the function $f_s : t \mapsto f(s, t)$ is not $\mu$-measurable is negligible (resp. locally negligible) for $\lambda$. b) The mapping $s \mapsto \int^\bullet f(s, t) \, d\mu(t)$ is $\lambda$-measurable, and

We begin by establishing b) when $f$ is $\nu$-moderated. By Lemma 2, this part of the statement is valid when there exists a compact subset $L$ of $T$ such that $f$ is zero outside $S \times L$; for, in this case $I_f = I_f^{L'}$ for every compact subset $L'$ of $T$ containing $L$, and formula (11) reduces to (6). In particular, b) is established for a function $f$ that is zero outside a compact subset of $S \times T$. On the other hand, Cor. 1 of Prop. 11 implies that b) is true when $f$ is $\nu$-negligible. Since every $\nu$-moderated function is the sum of a $\nu$-negligible function and a sequence of functions with compact support (§ 1, No. 9, Cor. 3 of Prop. 14), the assertion b) is true when $f$ is $\nu$-moderated.

Similarly, the assertion b) is obvious when $\mu$ is carried by a compact subset $L$ of $T$ (Lemma 2). Suppose that $\mu$ is moderated; then there exists a sequence $(\mu_n)_{n \in \mathbf{N}}$ of measures on $T$ with compact support, such that $\mu = \sum_n \mu_n$ (§ 1, No. 9, Cor. 5 of Prop. 14), whence $\nu = \sum_n \lambda \otimes \mu_n$ (No. 5, Cor. 3 of Prop. 10). The assertion b), being valid for each of the measures $\nu_n = \lambda \otimes \mu_n$, is also valid for $\nu = \sum_n \nu_n$.

Let us prove a); denote by $N$ the set of $s \in S$ such that $f_s$ is not $\mu$-measurable; for every compact subset $L$ of $T$, similarly denote by $N_L$ the set of $s \in S$ such that $f_s \varphi_L$ is not $\mu$-measurable. If $K$ and $L$ are compact sets in $S$ and $T$ respectively, then $f_{K \times L}$ is measurable with respect to the measure $\nu_{K \times L} = \lambda_K \otimes \mu_L$, and Prop. 2 of Ch. V, § 8, No. 2 shows that the set $N_L$ is locally negligible for $\lambda_K$; since $K$ is arbitrary, $N_L$ is locally $\lambda$-negligible.

Suppose that $f$ is zero outside a compact set of the form $K \times L$; then $N = N_L$, and $N$ is contained in $K$; it follows that $N$ is $\lambda$-negligible. Similarly, if $f$ is $\nu$-negligible, Cor. 2 of Prop. 11 implies that $N$ is $\lambda$-negligible. The case that $f$ is $\nu$-moderated can then be treated as above, on combining the preceding two cases.

Suppose that $\mu$ is carried by a compact subset $L$ of $T$; then $N = N_L$ again, therefore $N$ is locally $\lambda$-negligible. Since every moderated measure is the sum of a sequence of measures with compact support ($§ 1$, No. 9, Cor. 5 of Prop. 14), this result extends at once to the case that $\mu$ is moderated, on using Prop. 8 of $§ 1$, No. 7.