Corollary

— Let $X_i, X'_i$ ($1 \leq i \leq n$), $Y, Y'$ be locally compact spaces; let $\varphi, \varphi'$ be continuous mappings of $X = \prod_i X_i$ into $Y$ and of $X' = \prod_i X'_i$ into $Y'$, respectively; let $f_i$ be continuous mappings of $X_i$ into $X'_i$ ($1 \leq i \leq n$) and $g$ a continuous mapping of $Y$ into $Y'$, such that $\varphi' \circ f = g \circ \varphi$, $f$ being the mapping of $X$ into $X'$ that is the product of the $f_i$. Let $\mu_i$ be measures given respectively on the $X_i$, all $\neq 0$. Then the following two assertions are equivalent: (i) $f_i$ is $\mu_i$-proper for all $i$, and the measures $f_i(|\mu_i|)$ are $\varphi'$-convolvable; (ii) the $\mu_i$ are $\varphi$-convolvable, and $g$ is proper for $*_{\varphi}(|\mu_i|)$.

Moreover, when these assertions are verified,

For, let $h = \varphi' \circ f = g \circ \varphi$. By Prop. 1, the conditions (i) and (ii) are each equivalent to the following condition: (iii) the $\mu_i$ are $h$-convolvable. When this is so,