  Using elementary probabilistic methods, in particular a variant of the Weak
Law of Large Numbers related to the Bernoulli distribution, we prove that for
every infinite compact spaces $K$ and $L$ the product $K\times L$ admits a
sequence $\langle\mu_n\colon n\in\mathbb{N}\rangle$ of normalized signed
measures 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 completely
constructive -- the measures $\mu_n$ are defined by an explicit simple formula.
We also show that this result generalizes the classical theorem of Cembranos
and 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
$c_0$.
