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 $f_n$ is a sequence of integrable functions on a measure space $(X,\Sigma,\mu)$ that converges almost everywhere to another integrable function $f$ , then $\int |f_n - f| \, d\mu \to 0$ if and only if $\int | f_n | \, d\mu \to \int | f | \, d\mu$ .{{cite book|author=David Williams|title=Probability with Martingales|url=https://archive.org/details/probabilitywithm00will_764|url-access=limited|publisher=Cambridge University Press|location=New York|year=1991|page=[https://archive.org/details/probabilitywithm00will_764/page/n68 55]}}

The proof is based fundamentally on an application of the triangle inequality and Fatou's lemma.{{Cite web |title=Scheffé's Lemma - ProofWiki |url=https://proofwiki.org/wiki/Scheff%C3%A9%27s_Lemma |access-date=2023-12-09 |website=proofwiki.org |language=en |archive-date=2023-12-09 |archive-url=https://web.archive.org/web/20231209231730/https://proofwiki.org/wiki/Scheff%C3%A9%27s_Lemma |url-status=live }}

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 $\mu$ -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.{{cite journal |last1=Scheffe |first1=Henry |title=A Useful Convergence Theorem for Probability Distributions |journal=The Annals of Mathematical Statistics |date=September 1947 |volume=18 |issue=3 |pages=434–438 |doi=10.1214/aoms/1177730390|doi-access=free }} The result is a special case of a theorem by Frigyes Riesz about convergence in L^p spaces published in 1928.{{cite journal|journal=Periodica Mathematica Hungarica|date=September 2010|volume=61|number=1–2|pages=225–229|title= Why the theorem of Scheffé should be rather called a theorem of Riesz|author=Norbert Kusolitsch|doi=10.1007/s10998-010-3225-6|citeseerx=10.1.1.537.853|s2cid=18234313}}

References

Category:Theorems in measure theory