Difference between revisions of "User:Matteo.focardi/sandbox"
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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]] | + | 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 |
| − | + | $\mu$-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]]]. | |
| − | + | Note that 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 | |
| − | + | (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 | |
| + | locally compact space $X$ valued into a metrizable space $Y$. The conclusion of Egorov's theorem | ||
| + | may be false if $Y$ is not metrizable. | ||
====References==== | ====References==== | ||
Revision as of 12:31, 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. Let $\mu$ be a $\sigma$-additive measure defined on a $\sigma$-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
$\mu$-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 [1].
Note that 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
(cf. 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 locally compact space $X$ valued into a metrizable space $Y$. The conclusion of Egorov's theorem may be false if $Y$ 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 |
Matteo.focardi/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matteo.focardi/sandbox&oldid=28507