Algebraic genericity of strict-order integrability

Abstract

We provide sharp conditions on a measure µ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space Lp (µ, X) (p ≥ 1) which are not integrable with order q for any q > p (or any q < p) contains, except for zero, large subspaces of Lp (µ, X). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many nonintegrable functions of order q can be obtained even on any nonempty open subset of X, assuming that X is a topological space and µ is a Borel measure on X satisfying appropriate properties.