Illinois Journal of Mathematics

Purification of measure-valued maps

Peter Loeb and Yeneng Sun

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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.

Article information

Illinois J. Math., Volume 50, Number 1-4 (2006), 747-762.

First available in Project Euclid: 12 November 2009

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Zentralblatt MATH identifier

Primary: 28E05: Nonstandard measure theory [See also 03H05, 26E35]
Secondary: 03H05: Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] 91A06: n-person games, n > 2


Loeb, Peter; Sun, Yeneng. Purification of measure-valued maps. Illinois J. Math. 50 (2006), no. 1-4, 747--762. doi:10.1215/ijm/1258059490.

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