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arXiv · 1207.3818

Large structures made of nowhere $L^p$ functions

Szymon Glab·Pedro L. Kaufmann·Leonardo Pellegrini
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Abstract

We say that a real-valued function $f$ defined on a positive Borel measure space $(X,μ)$ is nowhere $q$-integrable if, for each nonvoid open subset $U$ of $X$, the restriction $f|_U$ is not in $L^q(U)$. When $(X,μ)$ 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-González, improve and complement recent spaceability and algebrability results from several authors and motivates new research directions in the field of spaceability.

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Szymon Glab, Pedro L. Kaufmann, Leonardo Pellegrini. 2012-10-30. Large structures made of nowhere $L^p$ functions. https://arxiv.org/abs/1207.3818

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