Therefore, the image of μ under the mapping x 7→ (x, T (x)) gives minimum to the integral on the right-hand side in (1). According to [4], if μ is absolutely continuous, the optimal transportation T is the gradient of a convex function V . As shown in [5], if μ = f dx and ν = g dx are two absolutely continuous probability measures such that μ is equivalent to Lebesgue measure and V is a convex ...

Given two metric spaces S1, S2 and a measurable function f : S1 → S2, sup­ pose S1 is equipped with some probability measure P. This induces a proba­ bility measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A)) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf−1 [X] is equal to EP[X(f)]. (Convince your­ self that this is the cas...

It is well known that a microperiodic function mapping a topological group into reals, which is continuous at some point is constant. We introduce the notion of a microperiodic multifunction, defined on a topological group with values in a metric space, and study regularity conditions implying an analogous result. We deal with Vietoris and Hausdorff continuity concepts.Stability of microperiodi...