Proposition 4

— Let $T$ and $T'$ be two locally compact spaces, $\mu$ (resp. $\mu'$) a positive measure on $T$ (resp. $T'$), $\nu = \mu \otimes \mu'$ the product measure on $X = T \times T'$. Let $f$ be a mapping of $X$ into $F$. Assume that: $1^\circ$ $f$ is scalarly $\nu$-integrable; $2^\circ$ there exists a locally $\mu$-negligible set $N \subset T$ such that, for every $t \notin N$, the mapping $t' \mapsto f(t, t')$ is scalarly $\mu'$-integrable, and $\int f(t, t') d\mu'(t') \in F$. Then, the function $t \mapsto \int f(t, t') d\mu'(t')$, defined for $t \notin N$, is scalarly essentially $\mu$-integrable and$^4$

$^3$It suffices to assume that the family is scalarly essentially $\mu$-integrable, thus the proposition holds for either of the two interpretations of ' $\mu$-adequate'. $^4$Equality as elements of ${F'}^*$.