A simple characterization of homogeneous Young measures and weak $L^1$ convergence of their densities

TL;DR

This paper provides a straightforward characterization of homogeneous Young measures linked to measurable functions, exploring their connection with weak convergence of measures and densities, including special cases involving smooth functions.

Contribution

It introduces a simple characterization of homogeneous Young measures using image measures and analyzes their weak* L1 convergence properties.

Findings

Homogeneous Young measures are characterized as constant mappings to probability measures.

Connections between weak convergence of measures and weak* L1 convergence of densities are established.

Special case analysis for smooth functions with Lebesgue-Stieltjes measures included.

Abstract

We formulate a simple characterization of homogeneous Young measures associated with measurable functions. It is based on the notion of the quasi-Young measure introduced in the previous article published in this Journal. First, homogeneous Young measures associated with the measurable functions are recognized as the constant mappings defined on the domain of the underlying function with values in the space of probability measures on the range of these functions. Then the characterization of homogeneous Young measures via image measures is formulated. Finally, we investigate the connections between weak convergence of the homogeneous Young measures understood as elements of the Banach space of scalar valued measures and the weak* L1 sequential convergence of their densities. A scalar case of the smooth functions and their Young measures being Lebesgue-Stieltjes measures is also analyzed.