Joint measurability and the one-way Fubini property for a continuum of independent random variables

Hammond, Peter; Sun, Yeneng

doi:10.1090/S0002-9939-05-08016-0

Joint measurability and the one-way Fubini property for a continuum of independent random variables
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by Peter J. Hammond and Yeneng Sun PDF

Proc. Amer. Math. Soc. 134 (2006), 737-747 Request permission

Abstract:

As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes an extension of the usual product measure-theoretic framework, using a natural “one-way Fubini” property. When the random variables are independent even in a very weak sense, this property guarantees joint measurability and defines a unique measure on a suitable minimal $\sigma$-algebra. However, a further extension to satisfy the usual (two-way) Fubini property, as in the case of Loeb product measures, may not be possible in general. Some applications are also given.

References

Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559
Robert M. Anderson, Nonstandard analysis with applications to economics, Handbook of mathematical economics, Vol. IV, Handbooks in Econom., vol. 1, North-Holland, Amsterdam, 1991, pp. 2145–2208. MR 1207198
Josef Berger, Horst Osswald, Yeneng Sun, and Jiang-Lun Wu, On nonstandard product measure spaces, Illinois J. Math. 46 (2002), no. 1, 319–330. MR 1936091
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
Donald L. Cohn, Measure theory, Birkhäuser, Boston, Mass., 1980. MR 578344, DOI 10.1007/978-1-4899-0399-0
J. L. Doob, Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Soc. 42 (1937), no. 1, 107–140. MR 1501916, DOI 10.1090/S0002-9947-1937-1501916-1
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
Richard Durrett, Probability, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Theory and examples. MR 1068527
Peter J. Hammond and Yeneng Sun, Monte Carlo simulation of macroeconomic risk with a continuum of agents: the symmetric case, Econom. Theory 21 (2003), no. 2-3, 743–766. Symposium in Honor of Mordecai Kurz (Stanford, CA, 2002). MR 2014352, DOI 10.1007/s00199-002-0302-y
Kenneth L. Judd, The law of large numbers with a continuum of IID random variables, J. Econom. Theory 35 (1985), no. 1, 19–25. MR 786985, DOI 10.1016/0022-0531(85)90059-6
H. Jerome Keisler, Hyperfinite model theory, Logic Colloquium 76 (Oxford, 1976) Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977, pp. 5–110. MR 0491155
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
Peter A. Loeb, A nonstandard functional approach to Fubini’s theorem, Proc. Amer. Math. Soc. 93 (1985), no. 2, 343–346. MR 770550, DOI 10.1090/S0002-9939-1985-0770550-9
Manfred Wolff and Peter A. Loeb (eds.), Nonstandard analysis for the working mathematician, Mathematics and its Applications, vol. 510, Kluwer Academic Publishers, Dordrecht, 2000. MR 1790871, DOI 10.1007/978-94-011-4168-0
H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
Yeneng Sun, Hyperfinite law of large numbers, Bull. Symbolic Logic 2 (1996), no. 2, 189–198. MR 1396854, DOI 10.2307/421109
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
Yeneng Sun, The almost equivalence of pairwise and mutual independence and the duality with exchangeability, Probab. Theory Related Fields 112 (1998), no. 3, 425–456. MR 1660898, DOI 10.1007/s004400050196
Sun, Y.N. The exact law of large numbers via Fubini extension and the characterization of insurable risks, J. Econ. Theory, published online, January 2005.