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In [[measure theory]], an area of [[mathematics]], '''Egorov's theorem''' establishes a condition for the [[uniform convergence]] of a [[pointwise convergence|pointwise convergent]] [[sequence]] of [[measurable function]]s. It is also named '''Severini–Egoroff theorem''' or '''Severini–Egorov theorem''', after [[Carlo Severini]], an [[Italia]]n [[mathematician]], and [[Dmitri Egorov]], a [[Russia]]n [[physicist]] and [[geometer]], who published independent proofs respectively in 1910 and 1911.

Egorov's theorem can be used along with [[support (mathematics)|compactly supported]] [[continuous function]]s to prove [[Lusin's theorem]] for [[integrable function]]s.

== Historical note ==
The first proof of the theorem was given by [[Carlo Severini]] in 1910 and was published in {{Harv|Severini|1910}}: he used the result as a tool in his research on [[Series (mathematics)|series]] of [[orthogonal functions]]. His work remained apparently unnoticed outside [[Italy]], probably due to the fact that it is written in [[Italian language|Italian]], appeared in a scientific journal with limited diffusion and was considered only as a means to obtain other theorems. A year later [[Dmitri Egorov]] published his independently proved results in the note {{Harv|Egoroff|1911}}, and the theorem become widely known under his name: however it is not uncommon to find references to this theorem as the Severini–Egoroff theorem or Severini–Egorov Theorem. According to {{Harvtxt|Cafiero|1959|p=315}} and {{Harvtxt|Saks|1937|p=17}}, the first mathematicians to prove independently the theorem in the nowadays common abstract [[measure space]] setting were [[Frigyes Riesz]] in {{Harv|Riesz|1922}}, {{Harv|Riesz|1928}}, and [[Wacław Sierpiński]] in {{Harv|Sierpiński|1928}}: an earlier generalization is due to [[Nikolai Luzin]], who succeeded in slightly relaxing the requirement of finiteness of measure of the [[Domain of a function|domain]] of convergence of the [[Pointwise convergence|pointwise converging functions]] in the ample paper {{Harv|Luzin|1916}}, as {{Harvtxt|Saks|1937|p=19}} recalls. Further generalizations were given much later by [[Pavel Korovkin]], in the paper {{Harv|Korovkin|1947}}, and by [[Gabriel Mokobodzki]] in the paper {{Harv|Mokobodzki|1970}}

== The formal statement of the theorem and its proof ==

===Statement of the theorem===
Let (''fn'') be a sequence of ''M''-valued measurable functions, where ''M'' is a separable metric space, on some [[measure space]] (''X'',Σ,μ), and suppose there is a [[measurable set|measurable subset]] ''A'' of finite μ-measure such that (''f''''n'') [[limit of a sequence|converges]] μ-[[almost everywhere]] on ''A'' to a limit function ''f''. The following result holds: for every ε > 0, there exists a measurable [[subset]] ''B'' of ''A'' such that μ(''B'') < ε, and (''fn'') [[uniform convergence|converges to ''f'' uniformly]] on the [[relative complement]] ''A'' \ ''B''.

Here, μ(''B'') denotes the μ-measure of ''B''. In words, the theorem says that pointwise convergence almost everywhere on ''A'' implies the apparently much stronger uniform convergence everywhere except on some subset ''B'' of arbitrarily small measure. This type of convergence is also called ''almost uniform convergence''.

===Discussion of assumptions and a counterexample===
*The hypothesis μ(''A'') < ∞ is necessary. To see this, it is simple to construct a counterexample when μ is the [[Lebesgue measure]]: consider the sequence of real-valued [[indicator function]]s

::<math>f_n(x) = 1_{[n,n+1]}(x),\qquad n\in\mathbb{N},\ x\in\mathbb{R},</math>

:defined on the [[real line]]. This sequence converges pointwise to the zero function everywhere but does not converge uniformly on ℝ \ ''B''  for any set ''B'' of finite measure: a counterexample in the general [[Dimension|<math>n</math>-dimensional]] [[real vector space]] ℝ''n'' can be constructed as shown by {{Harvtxt|Cafiero|1959|p=302}}.

*The separability of the metric space is needed to make sure that for ''M''-valued, measurable functions ''f'' and ''g'', the distance ''d''(''f''(''x''), ''g''(''x'')) is again a measurable real-valued function of ''x''.

===Proof===
For natural numbers ''n'' and ''k'', define the set ''En,k'' by the [[union (set theory)|union]]

:<math> E_{n,k} = \bigcup_{m\ge n} \left\{ x\in A \,\Big|\, |f_m(x) - f(x)| \ge \frac1k \right\}.</math>

These sets get smaller as ''n'' increases, meaning that ''E''''n''+1,''k'' is always a subset of ''En,k'', because the first union involves fewer sets. A point ''x'', for which the sequence (''fm''(''x'')) converges to ''f''(''x''), cannot be in every ''En,k'' for a fixed ''k'', because ''fm''(''x'') has to stay closer to ''f''(''x'') than 1/''k'' eventually. Hence by the assumption of μ-almost everywhere pointwise convergence on ''A'',

:<math>\mu\biggl(\bigcap_{n\in\mathbb{N}}E_{n,k}\biggr)=0</math>

for every ''k''. Since ''A'' is of finite measure, we have continuity from above; hence there exists, for each ''k'', some natural number ''nk'' such that

:<math>\mu(E_{n_k,k}) < \frac\varepsilon{2^k}.</math>

For ''x'' in this set we consider the speed of approach into the 1/''k''-[[neighbourhood (mathematics)|neighbourhood]] of ''f''(''x'') as too slow. Define

:<math> B = \bigcup_{k\in\mathbb{N}} E_{n_k,k}</math>

as the set of all those points ''x'' in ''A'', for which the speed of approach into at least one of these 1/''k''-neighbourhoods of ''f''(''x'') is too slow. On the set difference ''A'' \ ''B'' we therefore have uniform convergence.

Appealing to the [[sigma additivity]] of μ and using the [[geometric series]], we get
:<math>\mu(B)
\le\sum_{k\in\mathbb{N}}\mu(E_{n_k,k})
< \sum_{k\in\mathbb{N}}\frac\varepsilon{2^k}
=\varepsilon.</math>

== Generalizations ==
=== Luzin's version ===
[[Nikolai Luzin]]'s generalization of the Severini–Egorov theorem is presented here according to {{Harvtxt|Saks|1937|p=19}}.
==== Statement ====
Under the same hypothesis of the abstract Severini–Egorov theorem suppose that ''A'' is the [[union (set theory)|union]] of a [[sequence]] of [[measurable set]]s of finite μ-measure, and (''fn'') is a given sequence of ''M''-valued measurable functions on some [[measure space]] (''X'',Σ,μ), such that (''f''''n'') [[limit of a sequence|converges]] μ-[[almost everywhere]] on ''A'' to a limit function ''f'', then ''A'' can be expressed as the union of a sequence of measurable sets ''H'', ''A1'', ''A2'',... such that μ(''H'') = 0 and (''fn'') converges to ''f'' uniformly on each set ''Ak''.

==== Proof ====
It is sufficient to consider the case in which the set ''A'' is itself of finite μ-measure: using this hypothesis and the standard Severini–Egorov theorem, it is possible to define by [[mathematical induction]] a sequence of sets {''Ak''}k=1,2,... such that
:<math>\mu\left(A\setminus\bigcup_{k=1}^{N} A_k\right)\leq\frac{1}{N}</math>
and such that (''fn'') converges to ''f'' uniformly on each set ''Ak'' for each ''k''. Choosing
:<math>H=A\setminus\bigcup_{k=1}^{\infty} A_k</math>
then obviously μ(''H'') = 0 and the theorem is proved.

=== Korovkin's version ===
The proof of the Korovkin version follows closely the version on {{Harvtxt|Kharazishvili|2000|pp=183–184}}, which however generalizes it to some extent by considering [[admissible functional]]s instead of [[Measure (mathematics)|non-negative measures]] and [[inequality (mathematics)|inequalities]] <math>\scriptstyle\leq</math> and <math>\scriptstyle\geq</math> respectively in conditions 1 and 2.

==== Statement ====
Let (''M'',''d'') denote a [[separable space|separable]] [[metric space]] and (''X'',Σ) a [[measurable space]]: consider a [[measurable set]] ''A'' and a [[Set (mathematics)|class]] <math>\scriptstyle\mathfrak{A}</math> containing ''A'' and its measurable [[subset]]s such that their [[countable]] in [[Union (set theory)|unions]] and [[Intersection (set theory)|intersections]] belong to the same class. Suppose there exists a [[Measure (mathematics)|non-negative measure]] μ such that μ(''A'') exists and
# μ(<math>\scriptstyle\cap</math>''An'')=<math>\scriptstyle\lim</math>μ(''An'') if ''A1''<math>\scriptstyle\supset</math>''A2''<math>\scriptstyle\supset</math>... with ''An''<math>\scriptstyle\in\mathfrak{A}</math> for all ''n''
# μ(<math>\scriptstyle\cup</math>''An'')=<math>\scriptstyle\lim</math>μ(''An'') if ''A1''<math>\scriptstyle\subset</math>''A2''<math>\scriptstyle\subset</math>... with <math>\scriptstyle\cup</math>''An''<math>\scriptstyle\in\mathfrak{A}</math>.
If (''f''''n'') is a sequence of M-valued measurable functions [[limit of a sequence|converging]] μ-[[almost everywhere]] on ''A''<math>\scriptstyle\in\mathfrak{A}</math> to a limit function ''f'', then there exists a [[subset]] ''A′'' of ''A'' such that 0 < μ(''A'') − μ(''A′'')<ε and where the convergence is also uniform.

==== Proof ====
Consider the [[Indexed family|indexed family of sets]] whose [[index set]] is the set of [[natural number]]s ''m''<math>\scriptstyle\in\mathbb{N}</math>, defined as follows:
:<math>A_{0,m}=\left\{x\in A|d(f_n(x),f(x)) \le 1\ \forall n\geq m\right\}</math>
Obiviously
:<math>A_{0,1}\subseteq A_{0,2}\subseteq A_{0,3}\subseteq\dots</math>
and
:<math>A=\bigcup_{m\in\mathbb{N}}A_{0,m}</math>
therefore there is a [[natural number]] ''m0'' such that putting ''A0,m0''=''A0'' the following relation holds true:
:<math>0\leq\mu(A)-\mu(A_0)\leq\varepsilon</math>
Using ''A0'' it is possible to define the following indexed family
:<math>A_{1,m}=\left\{x\in A_0\left| d(f_m(x),f(x)) \le \frac12 \ \forall n\geq m\right.\right\}</math>
satifying the following two relationships, analogous to the previously found ones, i.e.
:<math>A_{1,1}\subseteq A_{1,2}\subseteq A_{1,3}\subseteq\dots</math>
and
:<math>A_0=\bigcup_{m\in\mathbb{N}}A_{1,m}</math>
This fact enable us to define the set ''A1,m1''=''A1'', where ''m1'' is a surely existing natural number such that
:<math>0\leq\mu(A)-\mu(A_1)\leq\varepsilon</math>
By iterating the shown construction, another indexed family of set {''An''} is defined such that it has the following properties:
*<math>\scriptstyle A_0\supseteq A_1\supseteq A_2\supseteq\dots</math>
*<math>\scriptstyle0\leq\mu(A)-\mu(A_m)\leq\varepsilon</math> for all ''m''<math>\scriptstyle\in\mathbb{N}</math>
*for each ''m''<math>\scriptstyle\in\mathbb{N}</math> there is a natural ''km'' such that for all ''n''<math>\scriptstyle\geq</math>''km'' then <math>\scriptstyle d(f_n(x),f(x)) \le 2^{-m}</math> for all ''x''<math>\scriptstyle\in</math>''Am''
and finally putting
:<math>A^\prime=\bigcup_{n\in\mathbb{N}}A_n</math>
the thesis is easily proved.

== Bibliography ==
*{{Citation
| last = Egoroff
| first = D. Th.
| authorlink = Dmitri Egorov
| title = Sur les suites des fonctions mesurables
| journal = [[Comptes rendus de l'Académie des sciences#1835-1965|Comptes rendus hebdomadaires des séances de l'Académie des sciences]]
| language = [[French language|French]]
| volume = 152
| pages = 244–246
| year = 1911
| url = http://gallica.bnf.fr/ark:/12148/bpt6k3105c/f244
| id =
| jfm = 42.0423.01
}}, available at [[Gallica]].
*{{Citation
| last = Riesz
| first = F.
| author-link = Frigyes Riesz
| title = Sur le théorème de M. Egoroff et sur les opérations fonctionnelles linéaires
| journal = [[Acta Scientiarum Mathematicarum|Acta Litt. Ac Sient. Univ. Hung. Francisco-Josephinae, Sec. Sci. Math.]] ([[Szeged]])
| language = [[French language|French]]
| volume = 1
| issue = 1
| pages = 18–26
| year = 1922
| url = http://acta.fyx.hu/acta/showCustomerArticle.action?id=4906&dataObjectType=article
| id =
| jfm = 48.1202.01
}}.
*{{Citation
| last = Riesz
| first = F.
| author-link = Frigyes Riesz
| title = Elementarer Beweis des Egoroffschen Satzes
| journal = [[Monatshefte für Mathematik und Physik]]
| language = [[German language|German]]
| volume = 35
| issue = 1
| pages = 243–248
| year = 1928
| url = http://www.springerlink.com/content/rk5r4p037225p542/?p=f3def8014bad41d5999fb3eb01a60ac6&pi=21
| doi = 10.1007/BF01707444
| id =
| jfm = 54.0271.04
}}.
*{{Citation
| last = Severini
| first = C.
| author-link = Carlo Severini
| title = Sulle successioni di funzioni ortogonali (On the sequences of orthogonal functions)
| journal = Atti dell'[http://www3.unict.it/gioenia/ Accademia Gioenia]
| language = [[Italian language|Italian]]
| series = serie 5a,
| volume = 3
| issue = 5
| pages = Memoria XIII, 1−7
| year = 1910
| id =
| jfm = 41.0475.04
}}.
*{{Citation
| last = Sierpiński
| first = W.
| author-link = Wacław Sierpiński
| title = Remarque sur le théorème de M. Egoroff
| journal = [[Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie]]
| language = [[French language|French]]
| volume = 21
| issue =
| pages = 84–87
| date =
| year = 1928
| url =
| id =
| jfm = 57.1391.03
}}.

== References ==
*{{Citation
| last = Beals
| first = Richard
| author-link =
| title = Analysis: An Introduction
| place = [[Cambridge]]
| publisher = [[Cambridge University Press]]
| year = 2004
| pages = x+261
| url = http://books.google.com/?id=cXAqJUYqXx0C&printsec=frontcover&dq=Analysis.+An+introduction.#v=onepage&q=
| id =
| mr = 2098699
| zbl = 1067.26001
| isbn = 0-521-60047-2
}}
*{{Citation
| last = Cafiero
| first = Federico
| author-link = Federico Cafiero
| title = Misura e integrazione
| place = [[Rome|Roma]]
| publisher = Edizioni Cremonese
| year = 1959
| series = Monografie matematiche del [[Consiglio Nazionale delle Ricerche]]
| volume = 5
| pages = VII+451
| id =
| mr = 0215954
| zbl = 0171.01503
| language = Italian
}}. ''Measure and integration'' (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of [[sequences]] of measure-related structures (measurable functions, [[measurable set]]s, measures and their combinations) is somewhat conclusive.
*{{MathWorld | author = Humphreys, Alexis |title=Egorov's Theorem |urlname=EgorovsTheorem}} Retrieved April 19, 2005.
*{{Citation
| last = Kharazishvili
| first = A.B.
| author-link =
| title = Strange functions in real analysis
| place = New York
| publisher = [[Marcel Dekker]]
| year = 2000
| series = Pure and Applied Mathematics – A Series of Monographs and Textbooks
| volume = 229
| edition = 1st
| pages = viii+297
| url = http://books.google.com/?id=8gHmxDSgcT0C&printsec=frontcover#v=onepage&q=
| doi =
| id =
| mr = 1748782
| zbl = 0942.26001
| isbn = 0-8247-0320-0}}. Contains a section named ''Egorov type theorems'', where the basic Severini–Egorov theorem is given in a form which slightly generalizes that of {{Harvtxt|Korovkin|1947}}.
*{{Citation
| last = Korovkin
| first = P.P.
| author-link =
| title = Generalization of a theorem of D.F. Egorov
| journal = [[Proceedings of the USSR Academy of Sciences|Doklady Akademii Nauk SSSR]]
| language = [[Russian language|Russian]]
| volume = 58
| issue =
| pages = 1265–1267
| year = 1947
| url =
| doi =
| id =
| mr = 0023322
| zbl = 0038.03803
}}
*{{Citation
| last = Luzin
| first = N.
| author-link = Nikolai Luzin
| title = Integral and trigonometric series
| journal = [[Matematicheskii Sbornik]]
| volume = 30
| issue = 1
| pages = 1–242
| year = 1916
| url = http://mi.mathnet.ru/eng/msb/v30/i1/p1
| jfm = 48.1368.01
| language = Russian
}}
*{{Citation
| last = Mokobodzki
| first = Gabriel
| author-link =
| title = Noyaux absolument mesurables et opérateurs nucléaires
| journal = [[Comptes rendus de l'Académie des sciences#1966-1980|Comptes rendus hebdomadaires des séances de l'Académie des sciences. Séries A]]
| language = French
| year = 1970
| volume = 270
| pages = 1673–1675
| date = 22 juin 1970
| url = http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f1683
| doi =
| id =
| mr = 0270182
| zbl = 0211.44803
}}
*{{Citation
| last = Saks
| first = Stanisław
| author-link = Stanisław Saks
| title = Theory of the Integral
| place = [[Warszawa]]-[[Lwów]]
| publisher = G.E. Stechert & Co.
| year = 1937
| series = [http://matwbn.icm.edu.pl/ksspis.php?wyd=10&jez=pl Monografie Matematyczne]
| volume = 7
| edition = 2nd
| pages = VI+347
| url = http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl
| id =
| jfm = 63.0183.05
| mr = 0017.30004
}} (available at the [http://matwbn.icm.edu.pl/ Polish Virtual Library of Science]). English translation by [[Laurence Chisholm Young]], with two additional notes by [[Stefan Banach]].

== External links ==
*{{PlanetMath|urlname=EgorovsTheorem|title=Egorov's theorem}}
*{{springer
| title= Egorov theorem
| id= E/e035120
| last=Kudryavtsev
| first= L.D.
| author-link=Lev Kudryavtsev
}}

[[Category:Theorems in measure theory]]
[[Category:Articles containing proofs]]

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