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.References
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Additional Information
- Bang-He Li
- Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- Email: libh@amss.ac.cn
- Tian-Hong Li
- Affiliation: Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- Email: thli@math.ac.cn
- Received by editor(s): June 14, 2010
- Received by editor(s) in revised form: November 3, 2010
- Published electronically: May 11, 2011
- Additional Notes: The first author is partially supported by the National Natural Science Foundation of China (NSFC) under Grant No. 10771206 and partially by 973 project (2004CB318000) of the People’s Republic of China.
The second author is supported by the Youth Foundation by Chinese NSF 10701073 and Chinese NSF 10931007, and is partially supported by the National Center for Theoretical Sciences and the National Tsing Hua University in Taiwan. Both authors thank the referee for many helpful comments. - Communicated by: Richard C. Bradley
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 207-215
- MSC (2010): Primary 28E05, 28A20, 26E35; Secondary 03H10, 49J45
- DOI: https://doi.org/10.1090/S0002-9939-2011-10903-1
- MathSciNet review: 2833533