Yegorov's theorem

Egorov's theorem is a theorem of measure theory , which shows the connection between point-wise convergence μ-almost everywhere and almost uniform convergence . Also part of the spellings find set of Egorov , set of Egorov or set of Egoroff attributable to a transfer of the name into English or French. The sentence is named after Dmitri Fjodorowitsch Jegorow , who proved it in 1911. The statement was already shown by Carlo Severini in 1910 , which is why it is named as a sentence by Egorov-Severini (or related spellings).

sentence

A finite dimensional space and measurable functions are given ${\displaystyle (X,{\mathcal {A}},\mu )}$

${\displaystyle f,(f_{n})_{n\in \mathbb {N} }\colon X\to \mathbb {K} }$.

If the sequence of functions converges pointwise μ-almost everywhere against , then it also converges almost uniformly against . ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle f}$

comment

Since the almost uniform convergence always leads to convergence almost everywhere, Jegorow's theorem provides the equivalence of the two types of convergence in the case of a finite measure space.

example

The following example shows that the statement is generally false for non-finite measure spaces. Looking at the sequence of functions

${\displaystyle f_{n}(x)=\chi _{[n,n+1]}(x)}$

on the measure space , this sequence of functions converges point by point (almost everywhere) to 0, because for anything is forever ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),\lambda )}$ ${\displaystyle x}$ ${\displaystyle n\geq \lceil x\rceil +1}$

${\displaystyle f_{n}(x)-0=\chi _{[n,n+1]}(x)-0=0}$.

But the sequence does not converge almost uniformly to 0, because is , then for every measurable set with measure smaller and each that , because has measure 1, cannot be contained in, and therefore ${\displaystyle 0<\varepsilon <1}$ ${\displaystyle A\subset \mathbb {R} }$ ${\displaystyle \varepsilon }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle [n,n+1]\setminus A\not =\emptyset }$ ${\displaystyle [n,n+1]}$ ${\displaystyle A}$

${\displaystyle \sup _{x\in \mathbb {R} \setminus A}|f_{n}(x)-0|=1}$

for all , that is, on no complement of a set of measure smaller, there can be uniform convergence. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \varepsilon }$

Original wording

In Jegorow's original work, the sentence was formulated only for functions on an interval:

Théorème - Si l'on a une suite de fonctions mesurables convergente pour tous les point d'un intervals AB sauf, peut-être, les points d'un ensemble de mesure nulle, on pourra tourjours enlever de l'intervalle AB un ensemble de mesure also petite qu'on voudra e tel que pour l'ensemble complémentaire [de mesure =  ] la suite est uniformément convergent. ${\displaystyle \eta }$ ${\displaystyle m(AB)-\eta }$

Translation: If one has a sequence of measurable functions that converges for all points of an interval AB, with the exception of possibly the points of a set of measure zero, then one can always get from the interval AB a set of measure that is as small as one is wants to remove, so that the sequence is  uniformly convergent on the complement set [with measure ]. ${\displaystyle \eta }$ ${\displaystyle m(AB)-\eta }$

The current term of almost uniform convergence was not yet in use. In the same paper, Jegorow suggested that this convergence, according to Hermann Weyl, be called essentially uniform .

Generalizations

Egorov's theorem also applies to measurable functions that take values ​​in a separable metric space .

See also

• Vectorial measure : for a generalization of the theorem for measures with values ​​in a Banach space

literature

• Isidor P. Natanson: Theory of the functions of a real variable. Unchanged reprint of the 4th edition. Harri Deutsch, Zurich et al. 1977, ISBN 3-87144-217-8 (also in digital form in Russian at INSTITUTE OF COMPUTATIONAL MODELING SB RAS, Krasnoyarsk ), Chapter IV., § 3.
• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .

Individual evidence

1. ↑ LD Kudryavtsev: Egorov theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
2. ↑ Elstrodt: dimensions and Integriationstheorie. 2009, p. 252.
3. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , sentence 3.1.3: Egoroff's theorem
4. ↑ D. Th. Egoroff: Sur les suites des fonctions mesurables : Comptes rendus 152 (1911), pages 244-246