Corollary 1

— Let $H$ be a subset of $T \times T'$, and let $A$ be the set of $t \in T$ such that the section $H(t)$ of $H$ at $t$ is not $\mu'$-negligible.

a) If $H$ is $\nu$-negligible then $A$ is $\mu$-negligible.

b) If $H$ is locally $\nu$-negligible and $\mu'$ is moderated, then $A$ is locally $\mu$-negligible.

The property a) follows at once from Prop. 7 (or from Prop. 6). Under the hypotheses of b), it comes to the same to say that $H(t)$ is locally μ′-negligible or that it is μ′-negligible, since μ′ is moderated (§1, No. 2, Prop. 7). Thus property b) follows from the formula (8).

This corollary extends at once, by passage to absolute values, to the product of two complex measures. The same is true for the following corollary: