Proposition 2

— Let $X_1, \ldots, X_n, Y$ be locally compact spaces, $\mu_i$ a bounded measure on $X_i$ ($1 \leq i \leq n$), $\mu$ the product of the $\mu_i$, $\varphi$ a $\mu$-measurable mapping of $\prod_{i=1}^n X_i$ into $Y$. Then the $\mu_i$ are $\varphi$-convolvable and

If the $\mu_i$ are moreover positive, then For, $\mu'_i = |\mu_i|$ is bounded and $\| \mu'_i \| = \| \mu_i \|$ (Ch. III, §1, No. 8, Cor. 1 of Prop. 10). One has $|\mu_1 \otimes \cdots \otimes \mu_n| = \mu'_1 \otimes \cdots \otimes \mu'_n$ (Ch. III, §4, Nos. 2, 4), therefore $\mu_1 \otimes \cdots \otimes \mu_n$ is bounded and (ibid., Prop. 4). Therefore $\varphi$ is $\mu$-proper (Ch. V, §6, No. 1, Remark 1), that is, the $\mu_i$ are $\varphi$-convolvable. One has $\| *_{i=1}^n \mu'_i \| = \| \mu'_1 \otimes \cdots \otimes \mu'_n \|$ (Ch. V, §6, No. 2, Th. 1), consequently $\| *_{i=1}^n \mu'_i \| = \| \mu'_1 \| \cdots \| \mu'_n \|$. Finally, $| *_{i} \mu_i | \leq *_{i} \mu'_i$ (No. 1, formula (2)), therefore