Large structures made of nowhere $L^p$ functions

TL;DR

This paper explores the structure of functions that are integrable in some regions but nowhere integrable in others, revealing large algebraic structures within these sets and advancing the understanding of spaceability in functional analysis.

Contribution

It introduces the concept of nowhere $q$-integrable functions and demonstrates their rich linear and algebraic structures, addressing open questions and extending previous spaceability results.

Findings

Sets of nowhere $q$-integrable functions contain large linear and algebraic structures.

Results answer an open question and improve existing spaceability theorems.

Motivates new research directions in the field of spaceability.

Abstract

We say that a real-valued function $f$ defined on a positive Borel measure space $(X,\mu)$ is nowhere $q$-integrable if, for each nonvoid open subset $U$ of $X$, the restriction $f|_U$ is not in $L^q(U)$. When $(X,\mu)$ satisfies some natural properties, we show that certain sets of functions defined in $X$ which are $p$-integrable for some $p$'s but nowhere $q$-integrable for some other $q$'s ($0<p,q<\infty$) admit a variety of large linear and algebraic structures within them. The presented results answer a question from Bernal-Gonz\'alez, improve and complement recent spaceability and algebrability results from several authors and motivates new research directions in the field of spaceability.