Properties of Lusin sets with applications to probability

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I. Notation. In this paper a Lusin set will be understood to be an uncountable subset of the interval I = [0, 1 ] such that if F is a first category subset of I, then Q F is, at most, a countable set or, equivalently, if { yi } is a countable dense subset of I and { Ei } is a sequence of positive numbers, then Q-UN(yi, Ei) is countable, where N(y, e) =In(y-e/2, y+E/2). By a universally measurable set we shall mean a subset E of R' such that if , is a nonatomic probability measure on the Borel subsets 63o of R', then ,u*(E) =,*(E). A set E is universally measurable if, and only if, every homeomorphism 4 of R' carries E onto a set q5(E) which is Lebesgue measurable. A universal null set is a universally measurable set which is mapped into a Lebesgue null set by all homneomorphisms. Since it imposes no real restriction to suppose that a probability measure on 6(B is nonatomic, henceforth we do so suppose. Finally, denote by (B the Borel subsets Q2lB, B-(Bo, of ?, i.e., 63=(3oI U.