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 ${\displaystyle f_{n}}$ is a sequence of integrable functions on a measure space ${\displaystyle (X,\Sigma ,\mu )}$ that converges almost everywhere to another integrable function ${\displaystyle f}$, then ${\displaystyle \int |f_{n}-f|\,d\mu \to 0}$ if and only if ${\displaystyle \int |f_{n}|\,d\mu \to \int |f|\,d\mu }$.^[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 ${\displaystyle \mu }$-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.^[3] The result is a special case of a theorem by Frigyes Riesz about convergence in L^p spaces published in 1928.^[4]

References

1. David Williams (1991). Probability with Martingales. New York: Cambridge University Press. p. 55.
2. "Scheffé's Lemma - ProofWiki". proofwiki.org. Archived from the original on 2023-12-09. Retrieved 2023-12-09.
3. 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.
4. 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. CiteSeerX 10.1.1.537.853. doi:10.1007/s10998-010-3225-6. S2CID 18234313.