Proposition 4

— Let $X_1, \ldots, X_n, Y$ be locally compact spaces, $\mu_i$ a measure on $X_i$ ($1 \leq i \leq n$), $S_i$ its support, and $\varphi$ a continuous mapping of $\prod_i X_i$ into $Y$ such that the restriction of $\varphi$ to $\prod_i S_i$ is proper. Then the $\mu_i$ are $\varphi$-convolvable.

For, let $K$ be a compact subset of $Y$. The support of $\mu = \mu_1 \otimes \cdots \otimes \mu_n$ is $S = \prod_i S_i$ (Ch. III, §4, No. 2, Prop. 2). Therefore $\overline{\varphi}(K) \cap \left( \prod_i X_i - S \right)$ is $\mu$-negligible. On the other hand, $\overline{\varphi}(K) \cap S$ is compact. Therefore $\overline{\varphi}(K)$ is $\mu$-integrable.