Monotone convergence theorem | Article about monotone convergence theorem by The Free Dictionary
monotone convergence theorem
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monotone convergence theorem[′män·ə‚tōn kən′vər·jəns ‚thir·əm]
The integral of the limit of a monotone increasing sequence of nonnegative measurable functions is equal to the limit of the integrals of the functions in the sequence.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Then the Monotone Convergence Theorem
gives that f is scalarly integrable, and
Notice also that, by the monotone convergence theorem
By using the definition of the Hilbert-Schmidt norm, the monotone convergence theorem
, Lemma 1 and formula (5), we obtain