In two recent papers, the
author studied extensions of several concepts of nonsmooth analysis to vector valued
operators. The purpose of the present work is to further continue this effort and to
study, from a probabilistic viewpoint, several properties of convex operators. In
particular, we will examine how various basic concepts of vectorial nonsmooth
analysis associated with an integrand f(ω,x) are related to those of the integral
operator F(x) =∫Ωf(ω,x)dμ(ω) where the vector valued integral is defined in the
sense of Bochner. Also we introduce a conditional expectation for such integrands,
study several of its properties, see how it is affected by various operations
of nonsmooth analysis, and derive a vector valued martingale convergence
theorem.
