A Bochner-type integration theory for random normed modules

TL;DR

This paper develops a measure and integration theory for random normed modules, extending classical theorems like Radon-Nikodým and Riesz-Markov-Kakutani to the setting of $L^0$-valued measures and modules.

Contribution

It introduces a Bochner-type integration framework for $L^0( ext{measure})$-valued measures on complete random normed modules, with several foundational theorems and applications.

Findings

Proved Radon-Nikodým theorem for $L^0$-valued measures.

Established Riesz-Markov-Kakutani representation theorem in this setting.

Outlined applications to martingales, random Radon-Nikodým property, and sets of finite perimeter.

Abstract

We develop a measure and integration theory for random normed modules. Given a probability space $({\rm X},\Sigma,\mathfrak m)$, we introduce and study measures taking values into the space $L^0(\mathfrak m)$ of $\mathfrak m$-measurable functions quotiented up to $\mathfrak m$-a.e. equality. Moreover, we develop a Bochner-type integration theory with respect to an $L^0(\mathfrak m)$-valued measure $\mu$, for maps whose target ${\rm M}$ is a complete random normed module with base $({\rm X},\Sigma,\mathfrak m)$, or equivalently an $L^0(\mathfrak m)$-Banach $L^0(\mathfrak m)$-module. Inter alia, we prove versions of the Radon-Nikod\'{y}m theorem and of the Riesz-Markov-Kakutani representation theorem for $L^0(\mathfrak m)$-valued measures. We also outline several applications of our integration theory: we introduce a notion of martingale with values in a complete random normed module, we propose a definition of random Radon-Nikod\'{y}m property and we discuss random sets of finite perimeter.