Abstract
We review some recent work by Yeneng Sun and the author. Sun’s work shows that there are results, some used for decades without a rigourous foundation, that arc only true for spaces with the rich structure of Loeb measure spaces. His joint work with the author uses that structure to extend an important result on the purification of measure valued maps.
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References
R.M. Anderson, “A nonstandard representation of Brownian motion and Itô integration”, Israel J. Math., 25 (1976) 15–46.
J.L. Doob. “Stochastic processes depending on a continuous parameter”, Trans. Amer. Math. Soc., 42 (1937) 107–140.
A. Dvoretsky, A. Wald and J. Wolfowitz, “Elimination of randomization in certain problems of statistics and of the theory of games”, Proc. Nat. Acad. Sci. USA, 36 (1950) 256–260.
A. Dvoretsky, A. Wald and J. Wolfowitz, “Relations among certain ranges of vector measures”, Pac. J. Math., 1 (1951) 59–74.
A. Dvoretsky, A. Wald and J. Wolfowitz, “Elimination of randomization in certain statistical decision problems in certain statistical decision procedures and zero-sum two-person games”, Ann. Math. Stat., 22 (1951) 1–21.
H.J. Keisler, An infinitesimal approach to stochastic analysis, Memoirs Amer. Math. Soc. 48, 1984.
H.J. Keisler, and Y.N. Sun, “A metric on probabilities, and products of Loeb spaces”, Jour. London Math. Soc., 69 (2004) 258–272.
M.A. Khan, K. P. Rath and Y. N. Sun, “The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games”, International Journal of Game Theory, 34 (2006) 91–104.
M.A. Khan and Y. N. Sun, “Non-cooperative games on hyperfinite Loeb spaces”, J. Math. Econ., 31 (1999) 455–492.
P.A. LOEB, “A combinatorial analog of Lyapunov’s Theorem for infinitesimally generated atomic vector measures”. Proc. Amer. Math. Soc., 39 (1973) 585–586.
P.A. Loeb, “Conversion from nonstandard to standard measure spaces and applications in probability theory”, Trans. Amer. Math. Soc., 211 (1975) 113–122.
P.A. Loeb and M. Wolee, eds., Nonstandard Analysis for the Working Mathematician. Kluwer Academic Publishers, Amsterdam, 2000.
P.A. Loeb, H. Osswald, Y. Sun and Z. Zhang, “Uncorrelatedness and orthogonality for vector-valued processes”, Tran. Amer. Math. Soc., 356 (2004) 3209–3225.
P.A. Loeb and Y. Sun, “Purification of measure-valued maps”, Doob Memorial Volume of the Illinois Journal of Mathematics, 50 (2006) 747–762.
H.L. Royden, Real Analysis, third edition, Macmillan, New York, 1988.
Y. Sun, “Hyperfinite law of large numbers”, Bull. Symbolic Logic, 2 (1996) 189–198.
Y. Sun, “A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN”, J. Math. Econ., 29 (1998) 419–503.
Y. Sun, “The almost equivalence of pairwise and mutual independence and the duality with exchangeability”, Probab. Theory and Relat. Fields, 112 (1998) 425–456.
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© 2007 Springer-Verlag Wien
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Loeb, P. (2007). Applications of rich measure spaces formed from nonstandard models. In: van den Berg, I., Neves, V. (eds) The Strength of Nonstandard Analysis. Springer, Vienna. https://doi.org/10.1007/978-3-211-49905-4_14
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DOI: https://doi.org/10.1007/978-3-211-49905-4_14
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