Suppose $X,Y$ to be compact spaces and $f\in C(X\times Y)$ different from zero. Is there two probability measures $\mu_f\in C(X)_+^*$, $\nu_f\in C(Y)_+^*$ such that $\mu_f\otimes\nu_f(|f|^2)>0$. In other words, do the product measure separate the elements of $C(X\times Y)$? If the answer is yes (as I suppose), I need a reference where such a proof is exhibited. If the answer is no, I need a counterexample.
2026-05-04 19:19:57.1777922397
do the product measures separate "points"?
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