invoking once again the Lebesgue

dominated convergence theorem, and passing to the limit as n [right arrow] +[infinity], we find that

Therefore, using the

dominated convergence theorem, the limit of the above integral as [epsilon] [right arrow] 0 is bounded by C' [H.

An application of the

dominated convergence theorem enables us to show that the function m [right arrow] Ew (X - [R.

5 (i), Vf (y, [[parallel][omega][parallel]) is bounded, we conclude by the continuity of f with respect to the second variable and by the

dominated convergence theorem, that

2] is uniformly bounded and tends to 0 pointwise, thus by the

dominated convergence theorem we obtain that (A5) holds for [[psi].

It turns out that Simons inequality [S] is a substitute to Lebesgue's

dominated convergence theorem when compactness and more generally topological regularity fails to hold, and it provides information on the behaviour of sequences as opposed to filters.

Consequently, by the Lebesgue

Dominated Convergence Theorem, [v.

The classical Riesz Representation Theorem and the Lebesgue

Dominated Convergence Theorem imply that every continuous linear real-valued functional C(X, R) [right arrow] R is represented by a unique regular Borel measure [mu] on X and, if [([f.

On the other hand, by using the

dominated convergence theorem and a density argument we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.