On linearity of pan-integral and pan-integrable functions space

TL;DR

This paper extends the $L^p$ space theory to the pan-integral under subadditive measures, proving linearity and completeness of the space, and connecting it to the concave integral and outer measures.

Contribution

It generalizes $L^p$ space theory to the pan-integral for subadditive measures, establishing linearity and Banach space structure.

Findings

Pan-integral is additive under subadditivity.

Pan-integrable functions form a Banach space.

Results apply to concave integrals and Lebesgue-like integrals from outer measures.

Abstract

$L\sp{p}$ space is a crucial aspect of classical measure theory. For nonadditive measure, it is known that $L\sp{p}$ space theory holds for the Choquet integral whenever the monotone measure $\mu$ is submodular and continuous from below. The main purpose of this paper is to generalize $L\sp{p}$ space theory to $+,\cdot$-based pan-integral. Let $(X, {\cal A}, \mu)$ be a monotone measure space. We prove that the $+,\cdot$-based pan-integral is additive with respect to integrands if $\mu$ is subadditive. Then we introduce the pan-integral for real-valued functions(not necessarily nonnegative), and prove that this integral possesses linearity if $\mu$ is subadditive. By using the linearity of pan-integral, we finally show that all of the pan-integrable functions form a Banach space. Since the $+,\cdot$-based pan-integral coincides with the concave integral for subadditive measure, the results obtained in this paper remain valid for the concave integral. Noticing that an outer measure is subadditive, we can define a Lebesgue-like integral(possesses linearity) from an outer measure, and the $L\sp{p}$ theory holds for this integral. {\it Keywords:} Monotone measure; Subadditivity; Pan-integral; Linearity; Pan-integrable space; Completeness