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2006 Purification of measure-valued maps
Peter Loeb, Yeneng Sun
Illinois J. Math. 50(1-4): 747-762 (2006). DOI: 10.1215/ijm/1258059490


Given a measurable mapping $f$ from a nonatomic Loeb probability space $(T,\mathcal{T},P)$ to the space of Borel probability measures on a compact metric space $A$, we show the existence of a measurable mapping $g$ from $(T,\mathcal{T},P)$ to $A$ itself such that $f$ and $g$ yield the same values for the integrals associated with a countable class of functions on $T\times A$. A corollary generalizes the classical result of Dvoretzky-Wald-Wolfowitz on purification of measure-valued maps with respect to a finite target space; the generalization holds when the domain is a nonatomic, vector-valued Loeb measure space and the target is a complete, separable metric space. A counterexample shows that the generalized result fails even for simple cases when the restriction of Loeb measures is removed. As an application, we obtain a strong purification for every mixed strategy profile in finite-player games with compact action spaces and diffuse and conditionally independent information.


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Peter Loeb. Yeneng Sun. "Purification of measure-valued maps." Illinois J. Math. 50 (1-4) 747 - 762, 2006.


Published: 2006
First available in Project Euclid: 12 November 2009

zbMATH: 1107.28007
MathSciNet: MR2247844
Digital Object Identifier: 10.1215/ijm/1258059490

Primary: 28E05
Secondary: 03H05, 91A06

Rights: Copyright © 2006 University of Illinois at Urbana-Champaign


Vol.50 • No. 1-4 • 2006
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