Distributional properties of correspondences on Loeb spaces
type: Item
instance of: scholarly article
title: Distributional properties of correspondences on Loeb spaces
zbMATH Open document ID: 0865.28008
author: Yeneng Sun
published in: Journal of Functional Analysis
publication date: 1996-12-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 28E05
Mathematics Subject Classification ID: 26E35
zbMATH DE Number: 944391
zbMATH Keywords: non-standard analysis
zbMATH Keywords: correspondence
zbMATH Keywords: selection
zbMATH Keywords: distribution
zbMATH Keywords: Loeb measure space
zbMATH Keywords: Young measure
zbMATH Keywords: non-standard models of probability
MaRDI profile type: Publication
OpenAlex ID: W2081762611
DOI: 10.1006/JFAN.1996.0079
reviewed by: Zvi Artstein
description: scientific article; zbMATH DE number 944391