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

Tom 221 / 2014

Szymon Głąb, Pedro L. Kaufmann, Leonardo Pellegrini Studia Mathematica 221 (2014), 13-34 MSC: Primary 46E30; Secondary 15A03. DOI: 10.4064/sm221-1-2

Streszczenie

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.

Autorzy

  • Szymon GłąbInstitute of Mathematics
    Technical University of Łódź
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail
  • Pedro L. KaufmannCAPES 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
    e-mail
  • Leonardo PellegriniInstituto de Matemática e Estatística
    Universidade de São Paulo
    Rua do Matão, 1010
    CEP 05508-900, São Paulo, Brazil
    e-mail

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