On Convergence of Conditional Expectation Operators

Abstract

Given an operator $T:U_X(\Sigma )\to Y$ or ${T:U(Σ)\to Y$, one may consider the net of conditional expectation operators $(T_{\pi })$ directed by refinement of the partitions $\pi$. It has been shown previously that $(T_{\pi })$ does not always converge to $T$. This paper gives several conditions under which this convergence does occur, including complete characterizations when $X=\mathbf{R}$ or when $X\sp *$ has the Radon-Nikodým property.