An example is given of a uniformly integrable stochastic sequence on a prob¬ability space (?Q, , P) which converges almost surely such that under successive con¬ditioning with respect to some sub-a-fields of % it is possible to switch from almost sure

convergence to non-convergence and vice versa. The example shows that in the dominated

convergence theorem for conditional expectations the assumption that the sequence is bounded in absolute value by an integrable random variable cannot be weakened or re¬placed bxr uniform inteorabilit