Date of this Version
The Annals of Statistics
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.
Brown, L. D., & Purves, R. (1973). Measurable Selections of Extrema. The Annals of Statistics, 1 (5), 902-912. http://dx.doi.org/10.1214/aos/1176342510
Date Posted: 27 November 2017
This document has been peer reviewed.