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

摘要

$L\sp{p}$ space is a crucial aspect of classical measure theory. Fornonadditive measure, it is known that $L\sp{p}$ space theory holds for theChoquet integral whenever the monotone measure $\mu$ is submodular andcontinuous 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 $+,\cdot$-basedpan-integral is additive with respect to integrands if $\mu$ is subadditive.Then we introduce the pan-integral for real-valued functions(not necessarilynonnegative), and prove that this integral possesses linearity if $\mu$ issubadditive. By using the linearity of pan-integral, we finally show that allof the pan-integrable functions form a Banach space. Since the $+,\cdot$-basedpan-integral coincides with the concave integral for subadditive measure, theresults obtained in this paper remain valid for the concave integral. Noticingthat an outer measure is subadditive, we can define a Lebesgue-likeintegral(possesses linearity) from an outer measure, and the $L\sp{p}$ theoryholds for this integral. {\it Keywords:} Monotone measure; Subadditivity; Pan-integral; Linearity;Pan-integrable space; Completeness