On Lebesgue points and measurability with Choquet integrals

TL;DR

This paper investigates Lebesgue points and measurability for non-negative functions with respect to Choquet integrals based on dyadic Hausdorff content, establishing convergence, density results, and sharpness examples.

Contribution

It develops a theory of Lebesgue points for non-measurable functions under Choquet integrals, including convergence and density theorems, with sharpness examples.

Findings

Convergence results for non-measurable functions with respect to Choquet integrals.

Density results between function spaces involving Choquet integrals.

Examples demonstrating the sharpness of the main convergence theorem.

Abstract

We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density result between function spaces. We provide examples which show sharpness of the main convergence theorem. These examples give additional information about the convergence in the norm also, namely the difference of the functions in this setting and continuous functions.