Random series in Lp(X, Σ,μ) using unconditional basic sequences and lp stable sequences: A result on almost sure almost everywhere convergence - Medina, Juan Miguel (2007)
Here we study the almost sure almost everywhere convergence of random series of the form Σ∞ i=1αifi in the Lebesgue spaces L p(X, Σ,μ), where the ai's are centered random variables, and the fi's constitute an unconditional basic sequence or an lp stable sequence. We show that if one of these series converges in the norm topology almost surely, then it converges almost everywhere almost surely.