A Bochner-type integration theory for random normed modules

We develop a measure and integration theory for random normed modules. Given a probability space $(\mathrm{X},\mathrm{\Sigma },\mathfrak{m})$, 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 $\mathrm{M}$ is a complete random normed module with base $(\mathrm{X},\mathrm{\Sigma },\mathfrak{m})$, 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.