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Difference between revisions of "Egorov theorem"

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====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</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)</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)</TD></TR></table>
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<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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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Revuz,  "Markov chains" , North-Holland  (1975)</TD></TR></table>
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<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 11:58, 27 September 2012

A theorem on the relation between the concepts of almost-everywhere convergence and uniform convergence of a sequence of functions. Let be a -additive measure defined on a -algebra , let , , and let a sequence of -measurable almost-everywhere finite functions , , converge almost-everywhere to a function . Then for any there exists a measurable set such that , and the sequence converges to uniformly on . For the case where is the Lebesgue measure on the line this was proved by D.F. Egorov [1].

Egorov's theorem has various generalizations extending its potentialities. For example, let be a sequence of measurable mappings of a locally compact space into a metrizable space for which the limit

exists locally almost-everywhere on with respect to a Radon measure . Then is measurable with respect to , and for any compact set and there is a compact set such that , and the restriction of to is continuous and converges to uniformly on . The conclusion of Egorov's theorem may be false if is not metrizable.

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


Comments

In 1970, G. Mokobodzki obtained a nice generalization of Egorov's theorem (see [a2], [a3]): Let , and be as above. Let be a set of -measurable finite functions that is compact in the topology of pointwise convergence. Then there is a sequence of disjoint sets belonging to such that the support of is contained in and such that, for every , the set of restrictions to of the elements of is compact in the topology of uniform convergence.

Egorov's theorem is related to the Luzin -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:
Egorov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Egorov_theorem&oldid=15906
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article