Scheffé's theorem

Scheffé's Lemma or Scheffé's Theorem is a convergence theorem of measure theory and statistics.

He says that from pointwise convergence almost everywhere of a sequence of functions and convergence in the p th means follows, if the sequence of L ^p norms converges. The set was shown by Frigyes Riesz in 1928 . It is usually formulated in textbooks for the case p = 1 and shown with an elegant approach by Phil Novinger from 1972 with the help of Fatou's lemma .

In 1947, Henry Scheffé independently proved that the almost certain convergence of a sequence of probability densities already results in the uniform convergence of the distributions. It can be shown that for such a sequence the uniform convergence of the distributions is equivalent to the L ¹ convergence of the densities and thus the problem is traced back to the already mentioned connection between convergence almost everywhere and convergence on average, which Scheffé about Lebesgues theorem received from the dominated convergence .

Formulation of the sentence for L ¹

Be . Then it follows from ${\displaystyle f,f_{1},\dotsc ,f_{n},\dotsc \in L^{1}(X,{\mathcal {A}},\mu )}$

${\displaystyle f_{n}\to f}$ ${\displaystyle \mu }$-Almost everywhere as well ${\displaystyle \int _{X}|f_{n}|\,\mathrm {d} \mu \to \int _{X}|f|\,\mathrm {d} \mu }$

beautiful

${\displaystyle \int _{X}|f_{n}-f|\,\mathrm {d} \mu \to 0}$.

literature

• Norbert Kusolitsch: Why the theorem of Scheffé should be rather called a theorem of Riesz. In: Periodica Mathematica Hungarica. Vol. 61, No. 1/2, 2010, ISSN  0031-5303 , pp. 225-229, doi : 10.1007 / s10998-010-3225-6 .
• W. Phil Novinger: Mean Convergence in L ^p Spaces. In: Proceedings of the American Mathematical Society. Vol. 34, No. 2, 1972, ISSN  0002-9939 , pp. 627-628, digital version (PDF; 134 kB) .
• Frédéric Riesz : Sur la convergence en moyenne. In: Acta Scientiarum Mathematicarum. Vol. 4, No. 1/2, 1928, ISSN  0001-6969 , pp. 58-64.
• Henry Scheffé: A Useful Convergence Theorem for Probability Distributions. In: Annals of Mathematical Statistics. Vol. 18, No. 3, 1947, ISSN  0003-4851 , pp. 434-438.