Scheffé’s lemma

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In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrals. 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.[1]


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 \muabsolutely continuous random variables implies convergence in distribution of those random variables.[1]


Henry Scheffé published a proof of the statement on convergence of probability densities in 1947. The result however is a special case of a theorem by Frigyes Riesz about convergence in Lp spaces published in 1928.[2]


  1. ^ a b David Williams (1991). Probability with Martingales. New York: Cambridge University Press. p. 55. 
  2. ^ Norbert Kusolitsch (September 2010). "Why the theorem of Scheffé should be rather called a theorem of Riesz". Periodica Mathematica Hungarica 61 (1-2): 225–229. doi:10.1007/s10998-010-3225-6.