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Young measures as probability distributions of Loeb spaces

Authors: Bang-He Li and Tian-Hong Li
Journal: Proc. Amer. Math. Soc. 140 (2012), 207-215
MSC (2010): Primary 28E05, 28A20, 26E35; Secondary 03H10, 49J45
Published electronically: May 11, 2011
MathSciNet review: 2833533
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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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  • 1. R. M. Anderson, Star-finite representations of measure space, Trans. Amer. Math. Soc. 271 (1982) 667-687. MR 654856 (83m:03077)
  • 2. L. Arkeryd, Loeb solutions of the Boltzmann equation, Arch. Ration. Mech. Anal. 86 (1984) 85-97. MR 748925 (85m:76047)
  • 3. E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim. 22 (1984) 570-598. MR 747970 (85k:49018)
  • 4. J. M. Ball, A version of the fundamental theorem for Young measures, Proc. Conf. on Partial Differential Equations and Continuum Models of Phase Transitions (Nice, 1988) (D. Serre, ed.), Springer, 1989. MR 1036070 (91c:49021)
  • 5. H. Berliocchi, J.M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France 101 (1973) 129-184. MR 0344980 (49:9719)
  • 6. G.-Q. Chen, Compactness Methods and Nonlinear Hyperbolic Conservation Laws, AMS/IP Studies in Advanced Mathematics, Vol. 15, Amer. Math. Soc., 2000, 33-75. MR 1767623 (2001f:35249)
  • 7. C. Castaing, P. Raynaud de Fitte, M. Valadier, Young measures on topological spaces. With applications in control theory and probability theory. Mathematics and Its Applications, 571. Kluwer Academic Publishers, Dordrecht, 2004. MR 2102261 (2005k:28001)
  • 8. N. J. Cutland, Internal controls and relaxed controls, J. London Math. Soc. (2) 27 (1983) 130-140. MR 686511 (84i:49019)
  • 9. N. J. Cutland, Loeb Measures in Practice: Recent Advances, Lecture Notes in Math., 1751, Springer-Verlag, 2000. MR 1810844 (2002k:28018)
  • 10. B. Dacorogna, Weak Continuity and Weak Lower Semi-continuity of Non-linear Functionals, Lecture Notes in Math., 922, Springer-Verlag, 1982. MR 658130 (84f:49020)
  • 11. N. Dunford, J. Schwartz, Linear Operators, Part I. General Theory, Interscience, New York, 1958. MR 0117523 (22:8302)
  • 12. B.-H. Li, On the moiré problem from distributional point of view, J. Sys. Sci. & Math. Sci., 6 (4) (1986) 263-268. MR 877260 (88i:03107)
  • 13. P. Lin, Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics, Trans. Amer. Math. Soc. 329 (1992) 377-413. MR 1049615 (92e:35106)
  • 14. P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975) 113-122. MR 0390154 (52:10980)
  • 15. E. McShane, Generalized curves, Duke Math. J. 6 (1940), 513-536. MR 0002469 (2:59d)
  • 16. E. McShane, Necessary conditions in generalized-curve problems of the calculus of variations, Duke Math. J. 7 (1940) 1-27. MR 0003478 (2:226a)
  • 17. H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. MR 0193469 (33:1689)
  • 18. L. Schwartz, Produits tensoriels topologiques d'espaces vectoriels topologiques, Espaces vectoriels topologiques nucléaires. Applications, Séminaire 1953/1954, Faculté des Sciences de Paris, 1954.
  • 19. K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, New York, 1976. MR 0491163 (58:10429)
  • 20. A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, Inc., 1963. MR 0098966 (20:5411)
  • 21. L. Tartar, Compensated compactness and applications to partial differential equations, Heriot-Watt Symposium, Vol. 4, Pitman, 1979. MR 0584398 (81m:35014)
  • 22. J. Warga, Relaxed variational problems, J. Math. Anal. Appl. 4 (1962) 111-128. MR 0142020 (25:5415a)
  • 23. J. Warga, Necessary conditions for minimum in relaxed variational problems, J. Math. Anal. Appl. 4 (1962) 129-145. MR 0142021 (25:5415b)
  • 24. J. Warga, Functions of relaxed controls, SIAM J. Control 5 (1967) 628-641. MR 0226474 (37:2064a)
  • 25. L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. Lett. Varsovie, Classe III, 30 (1937) 212-234.
  • 26. L. C. Young, Generalized surfaces in the calculus of variations. I, Ann. of Math. (2) 43 (1942) 84-103. MR 0006023 (3:249a)
  • 27. L. C. Young, Generalized surfaces in the calculus of variations. II, Ann. of Math. (2) 43 (1942) 530-544. MR 0006832 (4:49d)

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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

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

Keywords: Young measure, representation, nonstandard analysis, weak-star convergence, Loeb measure
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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