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 ${\displaystyle f_{n}}$ 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 ${\displaystyle \int |f_{n}|\,d\mu \to \int |f|\,d\mu }$.

The proof is based fundamentally on an application of the triangle inequality and Fatou's lemma. In probability theory, almost sure convergence can be weakened to requiring only convergence in probability.