Scheffé's lemma explained
In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrable functions. 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
\int|fn|d\mu\to\int|f|d\mu
.
[1] The proof is based fundamentally on an application of the triangle inequality and Fatou's lemma.[2]
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
-
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.[3] The result is a special case of a theorem by Frigyes Riesz about convergence in Lp spaces published in 1928.[4]
Notes and References
- Book: David Williams. Probability with Martingales. limited. Cambridge University Press. New York. 1991. 55.
- Web site: Scheffé's Lemma - ProofWiki . 2023-12-09 . proofwiki.org . en . 2023-12-09 . https://web.archive.org/web/20231209231730/https://proofwiki.org/wiki/Scheff%C3%A9%27s_Lemma . live .
- Scheffe . Henry . A Useful Convergence Theorem for Probability Distributions . The Annals of Mathematical Statistics . September 1947 . 18 . 3 . 434–438 . 10.1214/aoms/1177730390. free .
- Periodica Mathematica Hungarica. September 2010. 61. 1–2. 225–229. Why the theorem of Scheffé should be rather called a theorem of Riesz. Norbert Kusolitsch. 10.1007/s10998-010-3225-6. 10.1.1.537.853. 18234313.
