On Convergence of Conditional Expectation Operators

Given an operator $T:U_X(\Sigma)\to Y$ or {T:U(\Sigma)\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 $X={\bold R}$ or when $X\sp *$ has the Radon-Nikod\'ym property.