Young measures as probability distributions of Loeb spaces

Li, Bang-He; Li, Tian-Hong

doi:10.1090/S0002-9939-2011-10903-1

Young measures as probability distributions of Loeb spaces
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by Bang-He Li and Tian-Hong Li PDF

Proc. Amer. Math. Soc. 140 (2012), 207-215 Request permission

Abstract:

First we give a simple proof for a folklore result, which we call the Fundamental Theorem of Young Measures in a general framework. In the second part, we deal with the explicit representation of Young measures on Euclidean spaces. Young measure is an abstract concept in the sense that when it is described, it needs all continuous functions $\phi (y)$ and all $L^1$ functions $f(x)$ in the realm of standard analysis. However, we found that in the framework of nonstandard analysis, Young measures at almost all points are proved to be probability distributions for some random variables on some Loeb spaces defined in the monads of those points. This means that we can describe this Young measure without using $f(x)$ and $\phi (y)$. This also leads to the concrete computation of Young measures.

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