The Fatou inequality revisited. — Variations on a theme by A. Dvoretzky

Abstract

Definition i.i: Let (Xn, n f ~) be a sequence of random variables and Y: ~ ~ be a mapping. Then Y is said to be a measurable cluster point, or simply a cluster point of the sequence (Xn, n C ~) if Y is ~measurable and for almost every ~ @ ~, Y(m) is a cluster point of the sequence (Xn(~) , n f ~). Equivalently, Y is a cluster point of (Xn, n 6 ~) if and only if Y is measurable and li___mm ]X n Y[ = 0 a.s. n We now recall two results that will be used later.