Radon product of Haar measure is a Haar measure

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?