Almost uniform vs. pointwise convergence from a linear point of view

TL;DR

This paper reviews different modes of convergence of measurable function sequences, highlighting algebraic structures and constructing large subspaces and algebras within specific convergence families.

Contribution

It proves the existence of large vector subspaces and algebras within families of sequences with specific convergence properties under natural assumptions.

Findings

Existence of large vector subspaces in sequences converging pointwise almost everywhere but not almost uniformly.

Existence of large algebras in sequences converging almost uniformly but not almost everywhere.

Analysis of the algebraic structure of families of measurable function sequences.

Abstract

A review of the state of the art of the comparison between any two different modes of convergence of sequences of measurable functions is carried out with focus on the algebraic structure of the families under analysis. As a complement of the amount of results obtained by several authors, it is proved, among other assertions and under natural assumptions, the existence of large vector subspaces as well as of large algebras contained in the family of the sequences of measurable functions converging to zero pointwise almost everywhere but not almost uniformly, and in the family of the sequences of measurable functions converging to zero almost uniformly but not uniformly almost everywhere.