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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 ${\displaystyle \int |f_{n}|\,d\mu \to \int |f|\,d\mu }$.

The proof is based fundamentally on an application of the triangle inequality and Fatou's lemma.

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. The result is a special case of a theorem by Frigyes Riesz about convergence in L spaces published in 1928.

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.

• Theorems in measure theory