Proposition 7

— Let $f$ be a $\nu$-measurable positive numerical function defined on $T \times T'$.

a) If $f$ is $\nu$-moderated, then the functions $t \mapsto \int^* f(t, t') d\mu'(t')$, $t' \mapsto \int^* f(t, t') d\mu(t)$ are measurable and moderated for $\mu$ and $\mu'$, respectively, and

(7)

b) If the measure $\mu'$ is moderated, then the function $t \mapsto \int^* f(t, t') d\mu'(t')$ is $\mu$-measurable, and

(8)

The assertion a), as well as the assertion b) when $\mu'$ is bounded, are consequences of Prop. 5 of §3, No. 2. To treat the case that $\mu'$ is moderated, let us represent $\mu'$ as a sum $\sum_{n \in \mathbf{N}} \mu'_n$ of a sequence of bounded measures (§ 2, No. 3, Prop. 4). The function $t \mapsto \int^* f(t, t') d\mu'_n(t')$ is then $\mu$-measurable, and

But $\mu \otimes \mu' = \sum_{n \in \mathbf{N}} (\mu \otimes \mu'_n)$ (Prop. 1); the assertion b) is then obtained by summing on $n$ (§ 2, No. 2, Prop. 1).