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

Abstract

This paper investigates the linearity and integrability of the (+,⋅)-based pan-integrals on subadditive monotone measure spaces. It is shown that all nonnegative pan-integrable functions form a convex cone and the restriction of the pan-integral to the convex cone is a positive homogeneous linear functional. We extend the pan-integral to the general real-valued measurable functions. The generalized pan-integrals are shown to be symmetric and fully homogeneous, and to remain additive for all pan-integrable functions. Thus for a subadditive monotone measure the generalized pan-integral is linear functional defined on the linear space which consists of all pan-integrable functions. We define a p-norm on the linear space consisting of all p-th order pan-integrable functions, and when the monotone measure μ is continuous we obtain a complete normed linear space Lpanp(X,μ) equipped with the p-norm, i.e., an analogue of classical Lebesgue space Lp.