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A theorem on the relation between the concepts of almost-everywhere convergence and uniform convergence of a sequence of functions. Let $\mu$ be a $\sigma$-additive measure defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351203.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351204.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351206.png" />, and let a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351207.png" />-measurable almost-everywhere finite functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512010.png" /> converge almost-everywhere to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512011.png" />. Then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512012.png" /> there exists a measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512014.png" />, and the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512015.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512016.png" /> uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512017.png" />. For the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512018.png" /> is the Lebesgue measure on the line this was proved by D.F. Egorov [[#References|[1]]].
 
  
Egorov's theorem has various generalizations extending its potentialities. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512019.png" /> be a sequence of measurable mappings of a locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512020.png" /> into a metrizable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512021.png" /> for which the limit
+
{{MSC|28A}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512022.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
exists locally almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512023.png" /> with respect to a [[Radon measure|Radon measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512024.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512025.png" /> is measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512026.png" />, and for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512028.png" /> there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512030.png" />, and the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512031.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512032.png" /> is continuous and converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512033.png" /> uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512034.png" />. The conclusion of Egorov's theorem may be false if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512035.png" /> is not metrizable.
+
{{TEX|done}}
  
====References====
+
A  theorem on the relation between the concepts of  almost-everywhere  convergence and uniform convergence of a sequence of functions. In  literature it is sometimes cited as Egorov-Severini's
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D.F. Egorov,  "Sur les suites de fonctions mesurables"  ''C.R. Acad. Sci. Paris'' , '''152'''  (1911)  pp. 244–246  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961)  (Translated from Russian{{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8  (Translated from French{{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}} </TD></TR></table>
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theorem since it was proved independently and almost contemporarily by the two authors (see
 +
refs. {{Cite|Ego}}, {{Cite|Sev}}).
 +
 
 +
Let  $\mu$ be a [[Set function|$\sigma$-additive measure]] defined on a set $X$ endowed with a [[Algebra of sets|$\sigma$-algebra]] ${\mathcal A}$,  i.e. $(X,{\mathcal A})$ is a [[Measurable space|measurable space]].  
 +
Let  $E\in{\mathcal A}$,  $\mu(E)<+\infty$, and let $f_k:E\to\mathbb{R}$  be a sequence of $\mu$-measurable functions converging  $\mu$-almost-everywhere to a function $f$. Then, for every  $\varepsilon>0$ there exists a measurable set $E_\varepsilon\subset E$ such that $\mu(E\setminus E_\varepsilon)<\varepsilon$, and the sequence $f_k$ converges to $f$ uniformly on $E_\varepsilon$.
 +
 
 +
The  result is in general false if the condition $\mu(E)<+\infty$ is dropped. Despite of this, Luzin noted that if $X$, ${\mathcal A}$,  $\mu$, $f_k$ and $f$ are as above, and $E\in{\mathcal A}$ is the  countable union of sets $E_n$ with finite measure, then there exist a  sequence
 +
$\{A_n\}\subset\mathcal{A}$ and $H\in{\mathcal  A}$with $\mu(H)=0$, such that $E=(\cup_nA_n)\cup H$, and $f_k$ converges uniformly to $f$ on each $A_n$.
 +
 
 +
A typical application is when $\mu$ is a positive [[Radon measure|Radon measure]] defined on a topological space $X$
 +
(cf. [[Measure in a topological vector space|Measure in a topological vector space]]) and $E$ is a compact set.
 +
The case of the Lebesgue measure on the line  was first proved by D.F. Egorov ({{Cite|Ego}}).
  
 +
Egorov's theorem has various generalizations. For instance, it works for sequences of measurable functions defined on a
 +
measure space [[Measure space|$(X,{\mathcal A},\mu)$]] with values into a separable metric space $Y$. The conclusion  of
 +
Egorov's theorem might be false if $Y$ is not metrizable.
  
 +
Another generalization is due to  G. Mokobodzki  (see  {{Cite|DeMe}}, {{Cite|Rev}}):
 +
Let  $\mu$, ${\mathcal A}$ and $E$ be as above, and let $U$ be a set of  $\mu$-measurable finite functions that is compact in the topology of  [[Pointwise  convergence|pointwise convergence]].
 +
Then there is a sequence $\{A_n\}$ of disjoint sets  belonging to ${\mathcal A}$ such that the
 +
support of $\mu$ is contained in $\cup_nA_n$ and such that,  for every $n$, the restrictions
 +
to $A_n$ of the elements of $U$ is compact in the topology of
 +
[[Uniform convergence|uniform convergence]].
  
====Comments====
 
In 1970, G. Mokobodzki obtained a nice generalization of Egorov's theorem (see [[#References|[a2]]], [[#References|[a3]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512038.png" /> be as above. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512039.png" /> be a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512040.png" />-measurable finite functions that is compact in the topology of [[Pointwise convergence|pointwise convergence]]. Then there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512041.png" /> of disjoint sets belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512042.png" /> such that the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512043.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512044.png" /> and such that, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512045.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512046.png" /> of restrictions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512047.png" /> of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512048.png" /> is compact in the topology of [[Uniform convergence|uniform convergence]].
 
  
Egorov's theorem is related to the [[Luzin-C-property|Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512049.png" />-property]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,   "Measure theory" , v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer,  "Probabilities and potential" , '''C''' , North-Holland (1988)  (Translated from French)  {{MR|0939365}} {{ZBL|0716.60001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Revuz,  "Markov chains" , North-Holland (1975)  {{MR|0415773}} {{ZBL|0332.60045}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bou}}||      N. Bourbaki, "Elements of mathematics. Integration", Addison-Wesley      (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}}      {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}}      {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}}      {{ZBL|1095.28001}} {{ZBL|0156.06001}}
 +
|-
 +
|valign="top"|{{Ref|DeMe}}||
 +
C. Dellacherie,   P.A. Meyer,  "Probabilities and potential" , '''C''' , North-Holland   (1988)  (Translated from French)  {{MR|0939365}} {{ZBL|0716.60001}}  
 +
|-
 +
|valign="top"|{{Ref|Ego}}|| D.F.  Egorov,    "Sur les suites de fonctions mesurables"  ''C.R. Acad. Sci.  Paris'' ,  '''152'''  (1911)  pp. 244–246  {{MR|}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
 +
|valign="top"|{{Ref|Sev}}||  A.N. Kolmogorov,  S.V. Fomin,  "Elements of  the theory of functions  and functional analysis" , '''1–2''' , Graylock  (1957–1961)    (Translated from Russian)  {{MR|1025126}} {{MR|0708717}}  {{MR|0630899}}  {{MR|0435771}} {{MR|0377444}} {{MR|0234241}}  {{MR|0215962}}  {{MR|0118796}} {{MR|1530727}} {{MR|0118795}}  {{MR|0085462}}  {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}}  {{ZBL|0501.46001}}  {{ZBL|0501.46002}} {{ZBL|0235.46001}}  {{ZBL|0103.08801}}
 +
|-
 +
|valign="top"|{{Ref|Rev}}|| D. Revuz,  "Markov chains" , North-Holland   (1975)  {{MR|0415773}} {{ZBL|0332.60045}}  
 +
|-
 +
|valign="top"|{{Ref|Sev}}||
 +
C.  Severini, "Sulle successioni di funzioni ortogonali" (Italian), Atti  Acc. Gioenia, (5) 3 10 S, (1910) pp. 1−7 {{ZBL|41.0475.04}}
 +
|-
 +
|}

Revision as of 16:54, 18 October 2012

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

A theorem on the relation between the concepts of almost-everywhere convergence and uniform convergence of a sequence of functions. In literature it is sometimes cited as Egorov-Severini's theorem since it was proved independently and almost contemporarily by the two authors (see refs. [Ego], [Sev]).

Let $\mu$ be a $\sigma$-additive measure defined on a set $X$ endowed with a $\sigma$-algebra ${\mathcal A}$, i.e. $(X,{\mathcal A})$ is a measurable space. Let $E\in{\mathcal A}$, $\mu(E)<+\infty$, and let $f_k:E\to\mathbb{R}$ be a sequence of $\mu$-measurable functions converging $\mu$-almost-everywhere to a function $f$. Then, for every $\varepsilon>0$ there exists a measurable set $E_\varepsilon\subset E$ such that $\mu(E\setminus E_\varepsilon)<\varepsilon$, and the sequence $f_k$ converges to $f$ uniformly on $E_\varepsilon$.

The result is in general false if the condition $\mu(E)<+\infty$ is dropped. Despite of this, Luzin noted that if $X$, ${\mathcal A}$, $\mu$, $f_k$ and $f$ are as above, and $E\in{\mathcal A}$ is the countable union of sets $E_n$ with finite measure, then there exist a sequence $\{A_n\}\subset\mathcal{A}$ and $H\in{\mathcal A}$, with $\mu(H)=0$, such that $E=(\cup_nA_n)\cup H$, and $f_k$ converges uniformly to $f$ on each $A_n$.

A typical application is when $\mu$ is a positive Radon measure defined on a topological space $X$ (cf. Measure in a topological vector space) and $E$ is a compact set. The case of the Lebesgue measure on the line was first proved by D.F. Egorov ([Ego]).

Egorov's theorem has various generalizations. For instance, it works for sequences of measurable functions defined on a measure space $(X,{\mathcal A},\mu)$ with values into a separable metric space $Y$. The conclusion of Egorov's theorem might be false if $Y$ is not metrizable.

Another generalization is due to G. Mokobodzki (see [DeMe], [Rev]): Let $\mu$, ${\mathcal A}$ and $E$ be as above, and let $U$ be a set of $\mu$-measurable finite functions that is compact in the topology of pointwise convergence. Then there is a sequence $\{A_n\}$ of disjoint sets belonging to ${\mathcal A}$ such that the support of $\mu$ is contained in $\cup_nA_n$ and such that, for every $n$, the restrictions to $A_n$ of the elements of $U$ is compact in the topology of uniform convergence.


References

[Bou] N. Bourbaki, "Elements of mathematics. Integration", Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DeMe]

C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French) MR0939365 Zbl 0716.60001

[Ego] D.F. Egorov, "Sur les suites de fonctions mesurables" C.R. Acad. Sci. Paris , 152 (1911) pp. 244–246
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Sev] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[Rev] D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045
[Sev]

C. Severini, "Sulle successioni di funzioni ortogonali" (Italian), Atti Acc. Gioenia, (5) 3 10 S, (1910) pp. 1−7 Zbl 41.0475.04

How to Cite This Entry:
Egorov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Egorov_theorem&oldid=28514
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article