Scheffé's lemma
In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrable functions. It states that, if [math]\displaystyle{ f_n }[/math] is a sequence of integrable functions on a measure space [math]\displaystyle{ (X,\Sigma,\mu) }[/math] that converges almost everywhere to another integrable function [math]\displaystyle{ f }[/math], then [math]\displaystyle{ \int |f_n - f| \, d\mu \to 0 }[/math] if and only if [math]\displaystyle{ \int | f_n | \, d\mu \to \int | f | \, d\mu }[/math].[1]
The proof is based fundamentally on an application of the triangle inequality and Fatou's lemma.[2] In probability theory, almost sure convergence can be weakened to requiring only convergence in probability.[3]
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 [math]\displaystyle{ \mu }[/math]-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.[4] The result is a special case of a theorem by Frigyes Riesz about convergence in Lp spaces published in 1928.[5]
References
- ↑ David Williams (1991). Probability with Martingales. New York: Cambridge University Press. p. 55. https://archive.org/details/probabilitywithm00will_764.
- ↑ "Scheffé's Lemma - ProofWiki" (in en). https://proofwiki.org/wiki/Scheff%C3%A9's_Lemma.
- ↑ "real analysis - Generalizing Scheffe's Lemma using only Convergence in Probability". https://math.stackexchange.com/questions/4401886/generalizing-scheffes-lemma-using-only-convergence-in-probability.
- ↑ Scheffe, Henry (September 1947). "A Useful Convergence Theorem for Probability Distributions". The Annals of Mathematical Statistics 18 (3): 434–438. doi:10.1214/aoms/1177730390.
- ↑ Norbert Kusolitsch (September 2010). "Why the theorem of Scheffé should be rather called a theorem of Riesz". Periodica Mathematica Hungarica 61 (1–2): 225–229. doi:10.1007/s10998-010-3225-6.
 |  |
