Scheffé’s lemma

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In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrals. It states that, if is a sequence of integrable functions on a measure space that converges almost everywhere to another integrable function , then if and only if .[1]


Applied to probability theory, Scheffe's theorem, in the form stated here, implies that almost everywhere pointwise convergence of the probability density functions of a sequence of absolutely continuous random variables implies convergence in distribution of those random variables.[1]


Henry Scheffé published a proof of the statement on convergence of probability densities in 1947. The result however is a special case of a theorem by Frigyes Riesz about convergence in Lp spaces published in 1928.[2]


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