The Dominated Convergence Theorem in its classical from states: If | X n | Y a.s., 1, Y < , and either

The Dominated Convergence Theorem in its classical from states: If |X_n| ³Ya.s., ³ 1, εY< ¥, and either

Or

Then

Finite.

In the framework of conditional expectations, we have shown that: If |X_n| £ Y a.s., n ³1, εY < 1, εY < ¥, and

With εX finite, then

For any s-field B Í A.

By means of an example, show that the convergence

Cannot be replaced by

And still conclude that

a.s. X, X