Corollary 2

Let $f$ and $f'$ be two functions with values in $\mathbf{C}$ or in $\overline{\mathbf{R}}$, defined on $T$ and $T'$, respectively, and essentially integrable (resp. integrable) for the measures $\mu$ and $\mu'$, respectively. The function $f \otimes f'$ is then essentially integrable (resp. integrable) for the measure $\mu \otimes \mu'$, and

When $f$ and $f'$ are positive, $f \otimes f'$ is measurable by Cor. 3 of Prop. 3, and the statement follows from formula (10) (resp. (11)) and the criterion for essential integrability (§ 1, No. 3, Prop. 9) (resp. the integrability criterion of Ch. IV, § 5, No. 6, Th. 5). The general case then follows immediately.

Corollary 2 extends at once to the product of two complex measures.