Suppose we have a family of compact topological groups $\{G_\alpha\}_{\alpha \in A}$, define $G=\Pi G_\alpha$, it's a compact toplogical group equipped with coordinatewise multiplication and product topology. For each $G_\alpha$, we are given a Haar measure $\mu_\alpha$, prove that the Radon product of $\mu_\alpha$'s is a Haar measure on $G$.

I got no idea how to prove it, since we only know the formulation of Radon product in finite coordinate projection, how do we prove for a general Borel set?


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