On general characterization of Young measures associated with Borel functions

TL;DR

This paper characterizes Young measures linked to Borel functions as probability distributions of f(U), with U uniform, and shows their density via simple functions, providing insights into their structure and applications.

Contribution

It establishes a new characterization of Young measures for Borel functions and demonstrates their density with simple functions, advancing understanding of their structure.

Findings

Young measures for Borel functions are probability distributions of f(U).

Young measures for simple functions are weak* dense among those for measurable functions.

Examples illustrate applications of the main characterization.

Abstract

We prove that the Young measure associated with a Borel function f is a probability distribution of the random variable f(U), where U has a uniform distribution on the domain of f. As an auxiliary result, the fact that Young measures associated with simple functions are weak* dense in the set of Young measures associated with measurable functions is proved. Finally some examples of specific applications of the main result are presented with comments.