On the independence of correspondences

Lin, Xiaoai

doi:10.1090/S0002-9939-00-05652-5

On the independence of correspondences
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Proc. Amer. Math. Soc. 129 (2001), 1329-1334 Request permission

Abstract:

An almost independent set-valued process on a Loeb product space is shown to be representable as the closure of a sequence of its selections which are almost independent themselves. This provides a Castaing type representation in terms of independent correspondences. Different definitions of independence for correspondences in the literature are also unified in a general setting.

References

Zvi Artstein, On dense univalued representations of multivalued maps, Rend. Circ. Mat. Palermo (2) 33 (1984), no. 3, 340–350. MR 779939, DOI 10.1007/BF02844498
Zvi Artstein and Sergiu Hart, Law of large numbers for random sets and allocation processes, Math. Oper. Res. 6 (1981), no. 4, 485–492. MR 703091, DOI 10.1287/moor.6.4.485
Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
M. Ali Khan and Yeneng Sun, Non-cooperative games on hyperfinite Loeb spaces, J. Math. Econom. 31 (1999), no. 4, 455–492. MR 1688400, DOI 10.1016/S0304-4068(98)00031-7
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
Gheorghe Popescu, Asymptotic behavior of random systems with complete connections. I, Stud. Cerc. Mat. 30 (1978), no. 1, 37–68 (Romanian, with English summary). MR 483112
G. Matheron, Random sets and integral geometry, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1975. With a foreword by Geoffrey S. Watson. MR 0385969
Yeneng Sun, A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN, J. Math. Econom. 29 (1998), no. 4, 419–503. MR 1627287, DOI 10.1016/S0304-4068(97)00036-0
Y. N. Sun, The almost equivalence of pairwise and mutual independence and the duality with exchangeability, Probability Theory and Related Fields 112 (1998), 425-456.
Y. N. Sun, The complete removal of individual uncertainty: multiple optimal choices and random exchange economies, Economic Theory 14 (1999), 507-544.
Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056