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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 [[Set function|$\sigma$-additive measure]]  defined on a [[Algebra of sets|$\sigma$-algebra]] ${\mathcal A}$, let $E\in{\mathcal A}$,  $\mu(E)<+\infty$, and let $f_kE\to\mathbb{R}$ be a sequence of $\mu$-measurable functions converging
+
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
$\mu$-almost-everywhere to a function $f$. Then for any $\varepsilon>0$ 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 case  of the Lebesgue  measure on the line  was first proved by D.F. Egorov [[#References|[1]]].
+
theorem since it was proved independently and almost contemporarily by the two authors (see
Note that the result is in general false if the measure $\mu$ is only $\sigma$-finite.
+
refs. [[#References|[1]]], [[#References|[4]]]).
A typical application is when $\mu$ is a positive [[Radon measure|Radon measure]] defined on
 
(cf. [[Measure in a topological vector space|Measure in a topological vector space]]) defined on
 
of a topological space $X$ and $E$ is a compact set.
 
  
Egorov's theorem has various generalizations. For instance, to measurable mappings defined on a  
+
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]].
locally compact space $X$ valued into a metrizable space $Y$. The conclusion  of Egorov's theorem  
+
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 $E_\varepsilon$.
may be false if $Y$ is not metrizable.
+
 
 +
The result is in general false if the measure $\mu$ is only $\sigma$-finite.
 +
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 [[#References|[1]]].
 +
 
 +
Egorov's theorem has various generalizations. For instance, it works for sequences of measurable functions defined on a measure space [[Measure space|$(X,\mu,{\mathcal A})$]] with values into  
 +
a separable metric space $Y$. The conclusion  of Egorov's theorem may be false if $Y$ is not  
 +
metrizable.
 +
 
 +
The following generalization of Egorov's theorem was observed by Luzin: if $(X,\mu,{\mathcal A})$,
 +
$f_k$ and $f$ are as above, and $A\in{\mathcal A}$ is $\sigma$-finite, then there exist a sequence $\{A_n\}\subset\mathcal{A}$ and $H\in{\mathcal A}$, with $\mu(H)=0$, such that
 +
$A=(\cup_nA_n)\cup H$, and $f_k$ converges uniformly to $f$ on each $A_n$.
  
 
====References====
 
====References====
<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>
+
<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>
 
+
<TR><TD  valign="top">[4]</TD> <TD valign="top"> C. Severini, "Sulle successioni di funzioni ortogonali" (Italian), Atti Acc. Gioenia, (5) 3 10 S, (1910) pp. 1−7 {{ZBL|41.0475.04}}</TD></TR></table>
  
  
 
====Comments====
 
====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]].
+
In  1970, G. Mokobodzki obtained a nice generalization of Egorov's theorem  (see [[#References|[a2]]], [[#References|[a3]]]): Let $\mu$, ${\mathcal A}$ and $E$ be as above. 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]].
  
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]].
+
Egorov's  theorem is related to the [[Luzin-C-property|Luzin ${\mathcal C}$-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>
 
<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>

Revision as of 13:43, 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. [1], [4]).

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 measure $\mu$ is only $\sigma$-finite. 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 [1].

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

The following generalization of Egorov's theorem was observed by Luzin: if $(X,\mu,{\mathcal A})$, $f_k$ and $f$ are as above, and $A\in{\mathcal A}$ is $\sigma$-finite, then there exist a sequence $\{A_n\}\subset\mathcal{A}$ and $H\in{\mathcal A}$, with $\mu(H)=0$, such that $A=(\cup_nA_n)\cup H$, and $f_k$ converges uniformly to $f$ on each $A_n$.

References

[1] D.F. Egorov, "Sur les suites de fonctions mesurables" C.R. Acad. Sci. Paris , 152 (1911) pp. 244–246
[2] 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
[3] 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
[4] C. Severini, "Sulle successioni di funzioni ortogonali" (Italian), Atti Acc. Gioenia, (5) 3 10 S, (1910) pp. 1−7 Zbl 41.0475.04


Comments

In 1970, G. Mokobodzki obtained a nice generalization of Egorov's theorem (see [a2], [a3]): Let $\mu$, ${\mathcal A}$ and $E$ be as above. 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.

Egorov's theorem is related to the Luzin ${\mathcal C}$-property.

References

[a1] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[a2] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French) MR0939365 Zbl 0716.60001
[a3] D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045
How to Cite This Entry:
Matteo.focardi/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matteo.focardi/sandbox&oldid=28508