Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Devroye and L. Györfi, Nonparametric Density Estimation. The L1-View, Wiley, New York, 1985.
L. Devroye and L. Györfi, Distribution and density estimation, Principles of Nonparametric Learning, L. Györfi (ed.), CISM Courses and Lectures 434, Springer-Verlag, Wien, 2002, 211–270.
P. Duren, Theory of HpSpaces, Academic Press, New York, 1970, 21–22.
E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965, 207–209.
J. E. Littlewood, Lectures on the Theory of Functions, Oxford University Press, Oxford, 1944, 34–35.
W. P. Novinger, Mean Convergence in Lp Spaces, Proc. Amer. Math. Soc, 34 (1972), 627–628.
F. Riesz, Sur la convergence en moyenne, Acta Sci. Math. (Szeged), 4 (1928), 58–64.
H. Scheffé, A Useful Convergence Theorem for Probability Distributions, Ann. Math. Statistics, 18 (1947), 434–438.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays
Rights and permissions
About this article
Cite this article
Kusolitsch, N. Why the theorem of Scheffé should be rather called a theorem of Riesz. Period Math Hung 61, 225–229 (2010). https://doi.org/10.1007/s10998-010-3225-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-010-3225-6