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A necessary and sufficient condition on a sequence ${\{{\mathcal{A}_{n}}\}_{n\in \mathbb{N}}}$ of σ-subalgebras which assures convergence almost everywhere of conditional expectations for functions in ${L^{\infty }}$ is given. It is proven that for $f\in {L^{\infty }}(\mathcal{A})$ \[ \mathsf{E}(f|{\mathcal{A}_{n}})\stackrel{a.e.}{\longrightarrow }\mathsf{E}(f|{\mathcal{A}_{\mu a.e.}}).\]