Periodica Mathematica Hungarica

, Volume 61, Issue 1, pp 225-229

First online:

Why the theorem of Scheffé should be rather called a theorem of Riesz

  • Norbert KusolitschAffiliated withInstitute of Statistics and Probability Theory, Technical University Wien Email author 

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In 1947 Henry Scheffé published a result which afterwards became known as Scheffé’s theorem, stating that the distributions of a sequence (f n ) of densities, which converge almost everywhere to a density f, converge uniformly to the distribution of f. But almost 20 years earlier Frigyes Riesz proved a sufficient condition for convergence in the p-th mean (p ≥ 1), wherefrom the Scheffé theorem is just a special case.

Key words and phrases

Sheffé theorem convergence in total variation characterization of L p -convergence

Mathematics subject classification number

01A60 28A20