A NOTE ON GENERALIZED EGOROV’S THEORE

Abstract

We prove that the following generalized version of Egorov’s theorem is independent from the ZFC axioms of the set theory. Let {fn}n∈ω, fn : 0,1→R, be a sequence of functions (not necessarily measurable) converging pointwise to zero for every x ∈ 0,1.Then for every ε>0, there are a set A ⊂ 0,1 of Lebesgue outer measure m∗ > 1−ε and a sequence of integers {nk}k∈ω with {fnk}k∈ω converging uniformly on A