Theorem 1 (Lebesgue–Fubini)

— *Let $f$ be a function defined on $T \times T'$, with values in a Banach space $F$ or in $\overline{\mathbf{R}}$; let $N$ be the set of $t \in T$ such that the function $t' \mapsto f(t,t')$ is not $\mu'$-integrable.

a) Suppose that $f$ is $\nu$-integrable; then $N$ is $\mu$-negligible, the function $t \mapsto \int f(t,t')\,d\mu'(t')$ (defined for $t \notin N$) is $\mu$-integrable, and*

b) Suppose that $f$ is essentially $\nu$-integrable, and that the measure $\mu'$ is moderated; then $N$ is locally $\mu$-negligible, the function $t \mapsto \int f(t, t')\, d\mu'(t')$ (defined for $t \notin N$) is essentially $\mu$-integrable, and (13) holds.

The assertion a) follows at once from Th. 1 of §3, No. 3. To establish b), let us denote by $g$ a $\nu$-integrable function equal to $f$ locally almost everywhere, and by $H$ the set of $(t, t')$ such that $f(t, t') \neq g(t, t')$. By Cor. 1 of Prop. 7, the section $H(t)$ is $\mu'$-negligible, except for $t \in T$ forming a locally $\mu$-negligible set. The result pertaining to $f$ may then be deduced from the statement a) applied to $g$.