A counterexample on the continuity of conditional expectations (Q1100809)

Martingae convergence theorems give the answers for some questions about the continuity of conditional expectations. This paper shows an example of a converging sequence (in a natural sense) of \(\sigma\)-algebras such that the corresponding sequence of conditional expectations does not converge a.e. The similar convergence in \(L^ p\) (1\(\leq p\leq \infty)\) holds (Fetter's theorem) and a simple proof of this theorem is given in this paper, too.