On Universal Measurability and Perfect Probability
Purdue University A subset E of the real numbers R is an element of the set 1 of universally measurable subsets of R if, and only if, It*(E) = /1*(E) for each probability measure p on the Borel subsets X of R. A subset E of R is an element of the sigma ideal X of universal null sets if, and only if, p *(E) = 0 whenever ,I is a nonatomic probability measure on X. The purpose of this note is to recount some properties of the sigma algebra / and its sigma ideal X. When dealing with a finite, nonnegative measure p on a sigma algebra 99 of subsets of a set X it suffices, for our purposes, to normalize and, hence, suppose that ,I is an element of the set ?4(9) of probability measures on f. An element ,u of ?(#) is said to be perfect if for each #-measurable functionf, there exists Be 4 such that B cf(X) and pu(f -(B)) = 1. If A is a subset of R, then 6A will denote the sigma algebra of Borel subsets of A. D. Blackwell [1] used