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] In probability theory, almost sure convergence can be weakened to requiring only convergence in probability.^[3]

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.^[4] The result is a special case of a theorem by Frigyes Riesz about convergence in L^p spaces published in 1928.^[5]

References