Large structures made of nowhere $L^p$ functions

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.