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