Large structures made of nowhere $L^{q}$ functions

Szymon Głąb ¹   ; Pedro L. Kaufmann ²   ; Leonardo Pellegrini ³

¹ Institute of Mathematics Technical University of Łódź Wólczańska 215 93-005 Łódź, Poland
² CAPES Foundation Ministry of Education of Brazil Brasília/DF 70040-020, Brazil and Institute de Mathématiques de Jussieu Université Pierre et Marie Curie 4 Place Jussieu 75005 Paris, France
³ Instituto de Matemática e Estatística Universidade de São Paulo Rua do Matão, 1010 CEP 05508-900, São Paulo, Brazil

Studia Mathematica, Tome 221 (2014) no. 1, pp. 13-34

Résumé

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.

Détail

DOI : 10.4064/sm221-1-2

Keywords: say real valued function defined positive borel measure space nowhere q integrable each nonvoid subset restriction has natural properties certain sets functions defined which p integrable nowhere q integrable other infty admit variety large linear algebraic structures within presented results answer question bernal gonz lez improve complement recent spaceability algebrability results several authors motivate research directions field spaceability