  We obtain results concerning the so-called factorization for the convergence
of random variables almost everywhere (almost surely or with probability one),
belonging to the classical Lebesgue-Riesz spaces and we extend these results to
the Grand Lebesgue Spaces. We also give exact estimates for the parameters
involved and provide several examples. We also show that the obtained estimates
are, in the general case, essentially non-improvable, of course up to a
multiplicative constant.
