Large structures made of nowhere $L^{q}$ functions
Volume 221 / 2014
Studia Mathematica 221 (2014), 13-34
MSC: Primary 46E30; Secondary 15A03.
DOI: 10.4064/sm221-1-2
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 $(X,\mu )$ has 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 of Bernal-González, improve and complement recent spaceability and algebrability results of several authors and motivate new research directions in the field of spaceability.
