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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