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September, 1973 Measurable Selections of Extrema
L. D. Brown, R. Purves
Ann. Statist. 1(5): 902-912 (September, 1973). DOI: 10.1214/aos/1176342510

Abstract

Let $f: X \times Y \rightarrow R$. We prove two theorems concerning the existence of a measurable function $\varphi$ such that $f(x, \varphi(x)) = \inf_y f(x,y)$. The first concerns Borel measurability and the second concerns absolute (or universal) measurability. These results are related to the existence of measurable projections of sets $S \subset X \times Y$. Among other applications these theorems can be applied to the problem of finding measurable Bayes procedures according to the usual procedure of minimizing the a posteriori risk. This application is described here and a counterexample is given in which a Borel measurable Bayes procedure fails to exist.

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L. D. Brown. R. Purves. "Measurable Selections of Extrema." Ann. Statist. 1 (5) 902 - 912, September, 1973. https://doi.org/10.1214/aos/1176342510

Information

Published: September, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0265.28003
MathSciNet: MR432846
Digital Object Identifier: 10.1214/aos/1176342510

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 5 • September, 1973
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