  It is shown that there exist a subsequence for which the multiple ergodic
averages of commuting invertible measure preserving transformations of a
Lebesgue probability space converge almost everywhere provided that the maps
are weakly mixing with an ergodic extra condition. The proof provides a example
of non-singular dynamical system for which the maximal ergodic inequality does
not hold. We further get that the non-singular strategy to solve the pointwise
convergence of the Furstenberg ergodic averages fails.
