A NOTE ON GENERALIZED EGOROV’S THEOREM

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 nec

essarily measurable) converging pointwise to zero for every x g 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.

The following question was asked by F. Di Biase in connection with some

problem related to the behaviour of bounded harmonic functions on the open

unit disc in R2 (see [3]):

Suppose that {fn}n∈ω, fn : 0, 1 → R, is a sequence of functions converging

pointwise to zero for each x ∈ 0, 1 . Is it true that for every ε > 0, there are a

set A ⊂ 0, 1 of outer measure m∗ > 1 − ε and a sequence {nk}k∈ω such that

{fnk }k∈ω converges uniformly on A?

Notice that, by the well-known Egorov’s theorem, the answer is positive, if

we assume that {fn}n∈ω is a sequence of measurable functions