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]
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.
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]