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 $f_{n}$ is a sequence of integrable functions on a measure space $(X,\Sigma ,\mu )$ that converges almost everywhere to another integrable function $f$, then $\int |f_{n}-f|\,d\mu \to 0$ if and only if $\int |f_{n}|\,d\mu \to \int |f|\,d\mu$.

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 $\mu$-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^p spaces published in 1928.