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

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.

The third part of this article, 'Homogeneous Young measures', forms the core of the article 'Remarks on homogeneous Young measures associated with Borel functions', submitted in ArXiv. Supplied with the material described above in the Abstract has appeared in Optimization. The current version is as the first version of the article sent to Optimization with Acknowledgements added