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

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 [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] 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. [ABSTRACT FROM AUTHOR]