Proposition 5

— Let $(G_\iota)_{\iota \in I}$ be a family of locally compact groups. For every $\iota \in I$ let $\mu_\iota$ be a left (resp. right) Haar measure on $G_\iota$. Assume that there exists a finite subset $J$ of $I$ such that, for every $\iota \in I - J$, $G_\iota$ is compact and $\mu_\iota(G_\iota) = 1$. Then the product measure $\bigotimes_{\iota \in I} \mu_\iota$ is a left (resp. right) Haar measure on $G = \prod_{\iota \in I} G_\iota$. If $x = (x_\iota) \in G$ then

For every finite subset $J$ of $I$, $\bigotimes_{\iota \in J} \mu_\iota$ is a left (resp. right) Haar measure on $\prod_{\iota \in J} G_\iota$, as follows immediately from the definitions. Therefore $\bigotimes_{\iota \in I} \mu_\iota$ is a left (resp. right) Haar measure on $G$ (Ch. III, §4, No. 6, Prop. 9). On the other hand, if the $\mu_\iota$ are left Haar measures then whence $\Delta_G(x) = \prod_{\iota \in I} \Delta_{G_\iota}(x_\iota)$.