Dominated Convergence Theorem - Bounded Convergence Theorem

One corollary to the dominated convergence theorem is the bounded convergence theorem, which states that if ƒ₁, ƒ₂, ƒ₃, … is a sequence of uniformly bounded real-valued measurable functions which converges pointwise on a bounded measure space (S, Σ, μ) (i.e. one in which μ(S) is finite) to a function ƒ, then the limit ƒ is an integrable function and

$\lim_{n\to\infty} \int_S{f_n\,d\mu} = \int_S{f\,d\mu}.$

Remark: The pointwise convergence and uniform boundedness of the sequence can be relaxed to hold only μ-almost everywhere, provided the measure space (S, Σ, μ) is complete or ƒ is chosen as a measurable function which agrees μ-almost everywhere with the μ-almost everywhere existing pointwise limit.

Read more about this topic: Dominated Convergence Theorem

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