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
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
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.