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Abstract

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.