View source for Egorov theorem


{{MSC|28A}}

[[Category:Classical measure theory]]

{{TEX|done}}

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. {{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]].



====References====
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|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}}
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|valign="top"|{{Ref|DeMe}}|| 
C.  Dellacherie,    P.A. Meyer,   "Probabilities and potential" , '''C''' ,  North-Holland   (1988)  (Translated from French)  {{MR|0939365}}  {{ZBL|0716.60001}} 
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|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|}} 
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|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}} 
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|valign="top"|{{Ref|Rev}}|| D. Revuz,   "Markov chains" , North-Holland   (1975)  {{MR|0415773}} {{ZBL|0332.60045}} 
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|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}}
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