Proposition 5

*Let $X_1, \ldots, X_n, Y$ be locally compact spaces, $\mu_i$ a measure on $X_i$ ($1 \leq i \leq n$), $\mu$ the product of the $\mu_i$, $\varphi$ a $\mu$-proper mapping of $\prod_i X_i$ into $Y$, and $S_i$ the support of $\mu_i$.

a) The support of $*_{i} \mu_i$ is contained in the closure of $\varphi \left( \prod_i S_i \right)$.

b) If $\varphi$ is continuous and the $\mu_i$ are positive, then the support of $*_{i} \mu_i$ is the closure of $\varphi \left( \prod_i S_i \right)$.

Let $S = \prod_i S_i$ be the support of $\mu$. The support of $*_{i} \mu_i$ is contained in $\overline{\varphi(S)}$ by Ch. V, §6, No. 2, Cor. 3 of Prop. 2. If $\varphi$ is continuous and the $\mu_i$ are positive, then the support of $*_{i} \mu_i$ is $\overline{\varphi(S)}$ (loc. cit., Cor. 4 of Prop. 2).