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Using elementary probabilistic methods, in particular a variant of the WeakLaw of Large Numbers related to the Bernoulli distribution, we prove that forevery infinite compact spaces $K$ and $L$ the product $K\times L$ admits asequence $\langle\mu_n\colon n\in\mathbb{N}\rangle$ of normalized signedmeasures with finite supports which converges to $0$ with respect to the weak*topology of the dual Banach space $C(K\times L)^*$. Our approach is completelyconstructive -- the measures $\mu_n$ are defined by an explicit simple formula.We also show that this result generalizes the classical theorem of Cembranosand Freniche which states that for every infinite compact spaces $K$ and $L$the Banach space $C(K\times L)$ contains a complemented copy of the space$c_0$.

On complementability of $c_0$ in spaces $C(K\times L)$