Exercise 8, § 8

Let $T$ be a locally compact space, $\mu$ a positive measure $\neq 0$ on $T$, $A$ a discrete space, $\lambda$ the measure on $A$ defined by mass +1 at each point of $A$.

a) For a mapping $f$ of $A \times T$ into a topological space to be $(\lambda \otimes \mu)$-measurable, it is necessary and sufficient that, for every $\alpha \in A$, the mapping $t \mapsto f(\alpha, t)$ be $\mu$-measurable.

b) For a function $f$ defined on $A \times T$, with values in $\overline{\mathbf{R}}$ or in a Banach space, to be $(\lambda \otimes \mu)$-integrable, it is necessary and sufficient that: $1^\circ$ except for a countable set of values of $\alpha \in A$, the function $t \mapsto f(\alpha, t)$ be identically zero; $2^\circ$ for every $\alpha \in A$, the function $t \mapsto f(\alpha, t)$ be $\mu$-integrable; $3^\circ$ $\sum_{\alpha \in A} \int |f(\alpha, t)| d\mu(t) < +\infty$. For $f$ to be essentially $(\lambda \otimes \mu)$-integrable, it necessary and sufficient that: $1^\circ$ for each $\alpha \in A$, the function $t \mapsto f(\alpha, t)$ be essentially $\mu$-integrable; $2^\circ$ $\sum_{\alpha \in A} \int |f(\alpha, t)| d\mu(t) < +\infty$; one then has

c) Let $X$ be a locally compact space, $(\alpha, t) \mapsto \rho_{\alpha, t}$ a family of positive measures on $X$ ($\alpha \in A$, $t \in T$). In order that the family $(\alpha, t) \mapsto \rho_{\alpha, t}$ be $(\mu \otimes \nu)$-adequate, it is necessary and sufficient that, for every $\alpha \in A$, the family $t \mapsto \rho_{\alpha, t}$ be $\mu$-adequate, and that the family of measures $(\int \rho_{\alpha, t}\, d\mu(t))_{\alpha \in A}$ be summable.