Undominated Sequences of Integrable Functions

Abstract

In this paper, we investigate to what extent the conclusion of the Lebesgue dominated convergence theorem holds if the assumption of dominance is dropped. Specifically, we study both topological and algebraic genericity of the family of all null sequences of functions that, being continuous on a locally compact space and integrable with respect to a given Borel measure in it, are not controlled by an integrable function.

Lineability, algebrability, and sequences of random variables

“…In a nutshell, Theorem 2.2, Theorem 2.4 and Theorem 2.5 show that, without a dominating integrable function, the family of sequences not fulfilling the Dominated Convergence Theorem (see [11]) is very large. We continue in this direction and derive additional results related to the interchangeability of the integral and the limit (also compare with Theorem 2 in [24]).…”