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In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. It is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. ==Statement of the theorem== Lebesgue's Dominated Convergence Theorem. Let be a sequence of real-valued measurable functions on a measure space . Suppose that the sequence converges pointwise to a function ''f'' and is dominated by some integrable function ''g'' in the sense that : for all numbers ''n'' in the index set of the sequence and all points ''x'' ∈ ''S''. Then ''f'' is integrable and : which also implies : Remark 1. The statement "''g'' is integrable" is meant in the sense of Lebesgue; that is : Remark 2. The convergence of the sequence and domination by ''g'' can be relaxed to hold only almost everywhere provided the measure space is complete or ''f'' is chosen as a measurable function which agrees everywhere with the everywhere existing pointwise limit. (These precautions are necessary, because otherwise there might exist a non-measurable subset of a set , hence ''f'' might not be measurable.) Remark 3. If μ(''S'') < ∞, the condition that there is a dominating integrable function ''g'' can be relaxed to uniform integrability of the sequence , see Vitali convergence theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dominated convergence theorem」の詳細全文を読む スポンサード リンク
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