Scheffé’s lemma - Alchetron, The Free Social Encyclopedia

In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrals. It states that, if f n is a sequence of integrable functions on a measure space ( X , Σ , μ ) that converges almost everywhere to another integrable function f , then ∫ | f n − f | d μ → 0 if and only if ∫ | f n | d μ → ∫ | f | d μ .

Applications

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 μ -absolutely continuous random variables implies convergence in distribution of those random variables.

History

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 L^p spaces published in 1928.

References

Scheffé’s lemma Wikipedia

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