# Scheffé’s lemma

In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrals. 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]

## 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 however is a special case of a theorem by Frigyes Riesz about convergence in Lp spaces published in 1928.[2]

## References

1. ^ David Williams (1991). Probability with Martingales. New York: Cambridge University Press. p. 55.
2. ^ 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. doi:10.1007/s10998-010-3225-6.