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Scheffé's lemma

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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 is a sequence of integrable functions on a measure space that converges almost everywhere to another integrable function , then if and only if .[1]

The proof is based fundamentally on an application of the triangle inequality and Fatou's lemma.[2]

Applications

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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

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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]

References

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  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.