Proposition 1

*Let $X_{ij}$ ($1 \leq i \leq m, 1 \leq j \leq n_i$), $Y_i$ ($1 \leq i \leq m$), and Z be locally compact spaces; for each i, let $\varphi_i$ be a mapping of $X_i = \prod_j X_{ij}$ into $Y_i$; let $\varphi$ be the product of the $\varphi_i$, mapping $X = \prod_i X_i$ into $Y = \prod_i Y_i$; let $\psi$ be a mapping of Y into Z.

(i) Let $\mu_{ij}$ be measures given, respectively, on the $X_{ij}$, such that, for each i, the $\mu_{ij}$ ($1 \leq j \leq n_i$) are $\varphi_i$-convolvable, and such that the measures $*| \mu_{ij} |$ are $\psi$-convolvable; then the $\mu_{ij}$, for $1 \leq i \leq m, 1 \leq j \leq n_i$, are $(\psi \circ \varphi)$-convolvable and

(ii) Assume $\psi$ and the $\varphi_i$ continuous, and let $\mu_{ij}$ be measures $\neq 0$ given, respectively, on the $X_{ij}$ and $(\psi \circ \varphi)$-convolvable; then, for each i, the $\mu_{ij}$ ($1 \leq j \leq n_i$) are $\varphi_i$-convolvable, the measures $*_j |\mu_{ij}|$ are $\psi$-convolvable, and the formula (3) holds.

It suffices to consider the case that all of the measures in question are $\geq 0$.

Let us place ourselves under the hypotheses of (i). The mapping $\varphi$ is proper for $\bigotimes_{i,j} \mu_{ij}$, and

(Ch. V, §8, No. 5, Prop. 11). The mapping $\psi \circ \varphi$ is proper for $\bigotimes_{i,j} \mu_{ij}$, and

(Ch. V, §6, Prop. 4). Therefore the $\mu_{ij}$ ($1 \leq i \leq m,\ 1 \leq j \leq n_i$) are $(\psi \circ \varphi)$-convolvable and formula (3) holds.

Let us place ourselves under the hypotheses of (ii). First of all, Lemma 2 proves that $\varphi$ is proper for $\bigotimes_{i,j} \mu_{ij}$. Lemma 1 then proves that for every $i$, $\varphi_i$ is proper for $\bigotimes_j \mu_{ij}$, and that

By Lemma 2, $\psi$ is proper for $\bigotimes_i (*_j \mu_{ij})$. Whence the proposition.