View source for Egorov theorem

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]]].

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

<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>

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.

====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>



====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====
<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>