A Convergence Property for Conditional Expectation

Abstract

Convergence properties are obtained for repeated applications of the operator $f \rightarrow |f - E(f)|$, where $E$ denotes conditional expectation. If, for example, $E$ is the integral with respect to a probability measure $P, f \in L^\infty(P)$ and $T(f) = |f - E(f)|$, then $T^n(f)$ converges to 0 in $L^\infty(P)$ and $\Sigma T^n(f)$ converges in $L^1(P)$.