In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions. The dominated convergence theorem does not hold for the Riemann integral because the limit of a sequence of Riemann-integrable functions is in many cases not Riemann-integrable. 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.
Read more about Dominated Convergence Theorem: Statement of The Theorem, Proof of The Theorem, Discussion of The Assumptions, Bounded Convergence Theorem, Dominated Convergence in -spaces (corollary), Extensions
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