Successive Conditional Expectations of an Integrable Function

Abstract

Rota [5] has shown recently that if {T _n } is a sequence of conditional expectation operators

$${S_n} = {T_0}{T_1} \ldots {T_{n - 1}}{T_n}{T_{n - 1}} \ldots {T_1}{T_0},$$

(1)

and X is a random variable such that

$$E\left( {|X|{{\log }^ + }|X|} \right) < \infty, $$

(2)

then the sequence {S _n X} converges almost everywhere to an integrable function.

Received April 30, 1962.