Modes of convergence in nets, counterexamples, and lineability

Abstract

In this work we continue with the ongoing search for what are often large algebraic structures of mathematical objects (functions, sequences, etc.) which enjoy certain special properties. This type of study belongs to the recent area of research known as lineability. On this occasion, and among several other results, we shall show that there are large algebraic structures within (i) the set of nets which are weakly convergent, but are not bounded, (ii) nets that are weakly convergent, but are not convergent in norm, or (iii) the set of nets of measurable functions which converge pointwise to a function that is not measurable and that are bounded in $[0,1]$.