Algebraic genericity of strict-order integrability

Luis Bernal-González¹

¹ Facultad de Matemáticas Universidad de Sevilla 41080 Sevilla, Spain

Studia Mathematica, Tome 199 (2010) no. 3, pp. 279-293

Résumé

We provide sharp conditions on a measure $\mu$ defined on a measurable space $X$ guaranteeing that the family of functions in the Lebesgue space $L^p(\mu ,X)$ $(p \ge 1)$ which are not $q$-integrable for any $q > p$ (or any $q p$) contains large subspaces of $L^p(\mu ,X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-$q$-integrable functions can even be obtained on any nonempty open subset of $X$, assuming that $X$ is a topological space and $\mu$ is a Borel measure on $X$ with appropriate properties.

Détail

DOI : 10.4064/sm199-3-5

Keywords: provide sharp conditions measure defined measurable space guaranteeing family functions lebesgue space which q integrable contains large subspaces without zero improves recent results due aron garc palmberg rez puglisi seoane shown many non q integrable functions even obtained nonempty subset assuming topological space borel measure appropriate properties