# Lineability and modes of convergence

## TL;DR

This paper investigates the existence of large linear structures within classes of sequences of measurable functions that converge under various modes, highlighting their algebraic sizes and extending previous results.

## Contribution

It provides new insights into the algebraic size of sequences with specific convergence properties, expanding the understanding of lineability in functional analysis.

## Key findings

- Large linear structures exist among sequences converging in measure but not pointwise.
- There are substantial algebraic families of sequences converging uniformly but not in measure.
- The study extends previous results on the algebraic size of convergence classes.

## Abstract

In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e.~pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in $L^1$-norm, are analyzed. These findings extend and complement a number of earlier results by several authors.