Measurable Selections of Extrema
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Statistics Papers
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Statistics and Probability
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Brown, Lawrence D
Purves, R.
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Let f : X × Y → R. We prove two theorems concerning the existence of a measurable function φ such that f (x,φ(x)) = infy 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 ⊂ X × 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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1973
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The Annals of Statistics
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At the time of publication, author Lawrence Brown was affiliated with Cornell University. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.