Dominated Convergence Theorem - Dominated Convergence in -spaces (corollary)

Dominated Convergence in -spaces (corollary)

Let be a measure space, a real number and a sequence of-measurable functions .

Assume the sequence converges -almost everywhere to an -measurable function, and is dominated by a, i.e., for every holds -almost everywhere.

Then all as well as are in and the sequence converges to in the sense of, i.e.: .

Idea of the proof: Apply the original theorem to the function sequence with the dominating function .

Read more about this topic:  Dominated Convergence Theorem

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