Anti-Fubini and pseudo-Fubini functions
Fubini's theorem asserts that if a measurable function of two variables is integrable with respect to the product measure, then the iterated integrals coincide. In other words, if the iterated integrals do not coincide, then the function is not integrable for the product measure. The converse of Fubini's theorem is not true, in general. So, there exist measurable functions of two variables which are not integrable but the iterated integrals exist as real numbers and share the same value. In this paper the authors investigate the topological and algebraic size of the following classes of functions: \begin{itemize} \item[(1)] Measurable, non-integrable functions of two variables such that their iterated integrals do not coincide. \item[(2)] Measurable, non-integrable functions of two variables whose iterated integrals exist as real numbers and share the same value. \end{itemize} The topological size is investigated by means of the notion of residuality and the algebraic size is explored using the concepts of lineability and spaceability. The measures considered in the paper are \(\sigma\)-finite and the class of functions satisfying (1) is denoted by \(\alpha\mathcal{F}\) (anti-Fubini) and the class of functions satisfying (2) is represented by \(p\mathcal{F}\) (pseudo-Fubini). Several results on the size of \(\alpha\mathcal{F}\) and \(p\mathcal{F}\) are presented in the paper. For instance, it is shown that \(\alpha\mathcal{F}\) is not \(2\)-lineable, which is specially interesting because this kind of result is not common in the literature.
scientific article; zbMATH DE number 7365225
scientific article