Convergence of Conditional Expectations in Banach Function Spaces

We give a necessary and sufficient condition that assures the convergence of conditional expectations in a Banach function space. This extends the result of A. Alonso and F. Brambila-Paz w J. Math. Anal. Appl. 221 1998., 161]176x. We also consider the convergence of conditional expectations in a weighted L p-spaces.