Successive Conditional Expectations of an Integrable Function

then the sequence {SnX} converges almost everywhere to an integrable function. Here an example is given showing that this result fails to hold if (2) is replaced by E \X\ < co. Moreover, it is shown that (2) is a necessary condition for the almost everywhere convergence of {SnX} for every {Tn} if the underlying probability space (Q, a, P) is rich enough to support independent identically distributed random variables Xx, X2, • • • such that Xi = X. The idea behind our approach has several other applications. One is that (2) is a necessary condition for