Proposition 4

— Let $A$ be a subset of $T$. If $A$ is locally $\mu$-negligible, then $A \times T'$ is locally $\nu$-negligible. Conversely, if $A \times T'$ is locally $\nu$-negligible and if $\mu' \neq 0$, then $A$ is locally $\mu$-negligible.

The first assertion follows from Cor. 1 of Prop. 10 of §6, No. 6. To establish the second assertion, let us take up again the notations in the proof of Prop. 3; $A \times T' = \mathrm{pr}_1^{-1}(A)$ is locally negligible for the measure $\nu_1$, therefore $A$ is locally negligible for $a\mu$ (§ 6, No. 2, Cor. of Prop. 2), whence the result since $a \neq 0$.

The preceding statement extends at once to the product of two complex measures (Ch. III, §4, No. 2, Prop. 3), as does the following corollary.