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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351201.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351202.png" />-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]]].
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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
 
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

Revision as of 11:26, 18 October 2012

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 -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=28505
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article