Remark

— 1) When the measure $\mu'$ is not moderated, it can happen that $f$ is essentially $\nu$-integrable and the function $t' \mapsto f(t, t')$ is not essentially $\mu'$-integrable for any value of $t \in T$ (§3, Exer. 4).

2) Let $\mu$ and $\mu'$ be two complex measures, and let $\nu = \mu \otimes \mu'$. If $f$ is $\nu$-integrable (in other words, $|\nu|$-integrable), then application of the theorem to the measures $|\mu|$ and $|\mu'|$, whose product is $|\nu|$ (Ch. III, §4, No. 2, Prop. 3), implies that $t' \mapsto f(t, t')$ is $\mu'$-integrable for $\mu$-almost every $t$. From this one deduces, on decomposing the measures $\mu$ and $\mu'$ as a linear combination of positive measures, that the statement of a) extends to complex measures. One can argue similarly for b).