Distributional properties of correspondences on Loeb spaces
Within a non-standard analysis model, the paper studies the ensemble \({\mathcal D}_F\) of distributions of selections \(f\) of a correspondence \(F\), defined on a measure space with values in a Polish space. (A correspondence is a set-valued map; a selection satisfies \(f(t)\in F(t)\) for all \(t\); the distribution of \(f\) is the induced probability on the range space.) Standard examples show that \({\mathcal D}_F\) might not be convex or compact (in the weak convergence) even when \(F\) is atomless, measurable, bounded, and has closed values. In the non-standard model, specifically in a Loeb measure space, the situation is different. The paper establishes the convexity and compactness of \({\mathcal D}_F\) and characterizes compactness in terms of tightness. Semi-continuity is established when \(F\) depends likewise on a parameter, and for a Young measure (or random measure, relaxed control) it is shown that its distribution is realized, in the non-standard model, by a selection from the support mapping. The derivations are accompanied by enlightening remarks on the role of the non-standardness, thus the paper is a nice addition to the literature promoting non-standard models of probability and their applications.
scientific article; zbMATH DE number 944391