Question

Give an example satisfying the following conditions: give out a sequence of random variables defined on a probability space, and a sub sigma algebra: the sequence converges almost surely to a limit and it is also uniformly integrable, but the sequence of the conditional expectation of the random variable sequence on the sub sigma algebra does not converges almost surely to the limit, which is the conditional expectation of the limit of the random variable sequence on the sub sigma algebra.

Answer 1

Consider independent random variables $X_n, Y_n, n\in\mathbb N$, such that $\mathbb P(X_n = 1) = 1-\mathbb P(X_n = 0) = 1/n$, $\mathbb P(Y_n = n) = 1-\mathbb P(Y_n = 0) = 1/n$. Set $Z_n = X_nY_n$. Then, $Z_n$ is uniformly integrable and $Z_n\to 0$ as $n\to\infty$ almost surely, by Borel-Cantelli Lemma. However, set ${\cal A} = \sigma(X_n:n\in\mathbb N)$. Then $\mathbb E(Z_n\mid{\cal A}) = X_n\mathbb E(Y_n\mid {\cal A}) = X_n$, which converges to 0 in $L^1$ but not almost surely.