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Vitali's covering theorem. If a system of closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967801.png" /> is a Vitali covering (see below) of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967802.png" />, it is possible to extract from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967803.png" /> an at most countable sequence of pairwise disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967805.png" /> such that
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Vitali's covering theorem. If a system of closed sets $\mathcal F$ is a Vitali covering (see below) of a set $A\subset\mathbb R^n$, it is possible to extract from $\mathcal F$ an at most countable sequence of pairwise disjoint sets $\{F_i\}$, $i=1,2,\dots$, such that
 +
\begin{equation}
 +
m_e\left[A\setminus\bigcup_{i=1}^{\infty}F_i\right]=0,
 +
\end{equation}
 +
where $m_e$ is the outer [[Lebesgue measure|Lebesgue measure]] in $\mathbb R^n$.
  
<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/v/v096/v096780/v0967806.png" /></td> </tr></table>
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A Vitali covering of a set $A\subset\mathbb R^n$ is a system $\mathcal E$ of subsets of $\mathbb R^n$ such that for any $x\in A$ there exists a sequence $\{E_n\}$ from $\mathcal E$ satisfying the following conditions:
 +
\begin{equation}
 +
x\in\bigcap_{n=1}^{\infty}E_n;
 +
\end{equation}
 +
\begin{equation}
 +
\delta_n = \delta(E_n) \to 0\quad \text{ if } n\to\infty,
 +
\end{equation}
 +
where $\delta(E_n)$ is the diameter of $E_n$; and
 +
\begin{equation}
 +
\inf_n\left[\sup\frac{m_e(E_n)}{m(I)}\right]=\alpha>0,
 +
\end{equation}
 +
where the supremum is taken over all $I$ (cubes with faces parallel to the coordinate planes and containing $E_n$), this supremum is said to be the regularity parameter of $E_n$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967807.png" /> is the outer [[Lebesgue measure|Lebesgue measure]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967808.png" />.
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The theorem was demonstrated by G. Vitali [[#References|[1]]] for the case when $\mathcal F$ consists of cubes with faces parallel to the coordinate planes. Vitali's theorem is valid as stated if $\mathcal F$ is a Vitali covering of the set $A$ and not for a covering in the ordinary sense. This condition must always be satisfied, even if $\mathcal F$ is a system of segments and if to each $x\in A$ there corresponds a sequence $\{F_n\}$ from $\mathcal F$ with centres at $x$ and with diameters tending to zero.
 
 
A Vitali covering of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v0967809.png" /> is a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678010.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678011.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678012.png" /> there exists a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678013.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678014.png" /> satisfying the following conditions:
 
 
 
<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/v/v096/v096780/v09678015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
 
 
 
<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/v/v096/v096780/v09678016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678017.png" /> is the diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678018.png" />; and
 
 
 
<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/v/v096/v096780/v09678019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
 
 
 
where the supremum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678020.png" /> (cubes with faces parallel to the coordinate planes and containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678021.png" />), and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678022.png" /> is the outer Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678023.png" />; this supremum is said to be the regularity parameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678024.png" />.
 
 
 
The theorem was demonstrated by G. Vitali [[#References|[1]]] for the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678025.png" /> consists of cubes with faces parallel to the coordinate planes. Vitali's theorem is valid as stated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678026.png" /> is a Vitali covering of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678027.png" /> and not for a covering in the ordinary sense. This condition must always be satisfied, even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678028.png" /> is a system of segments and if to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678029.png" /> there corresponds a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678030.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678031.png" /> with centres at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678032.png" /> and with diameters tending to zero.
 
  
 
====References====
 
====References====

Revision as of 07:55, 11 December 2012

Vitali's covering theorem. If a system of closed sets $\mathcal F$ is a Vitali covering (see below) of a set $A\subset\mathbb R^n$, it is possible to extract from $\mathcal F$ an at most countable sequence of pairwise disjoint sets $\{F_i\}$, $i=1,2,\dots$, such that \begin{equation} m_e\left[A\setminus\bigcup_{i=1}^{\infty}F_i\right]=0, \end{equation} where $m_e$ is the outer Lebesgue measure in $\mathbb R^n$.

A Vitali covering of a set $A\subset\mathbb R^n$ is a system $\mathcal E$ of subsets of $\mathbb R^n$ such that for any $x\in A$ there exists a sequence $\{E_n\}$ from $\mathcal E$ satisfying the following conditions: \begin{equation} x\in\bigcap_{n=1}^{\infty}E_n; \end{equation} \begin{equation} \delta_n = \delta(E_n) \to 0\quad \text{ if } n\to\infty, \end{equation} where $\delta(E_n)$ is the diameter of $E_n$; and \begin{equation} \inf_n\left[\sup\frac{m_e(E_n)}{m(I)}\right]=\alpha>0, \end{equation} where the supremum is taken over all $I$ (cubes with faces parallel to the coordinate planes and containing $E_n$), this supremum is said to be the regularity parameter of $E_n$.

The theorem was demonstrated by G. Vitali [1] for the case when $\mathcal F$ consists of cubes with faces parallel to the coordinate planes. Vitali's theorem is valid as stated if $\mathcal F$ is a Vitali covering of the set $A$ and not for a covering in the ordinary sense. This condition must always be satisfied, even if $\mathcal F$ is a system of segments and if to each $x\in A$ there corresponds a sequence $\{F_n\}$ from $\mathcal F$ with centres at $x$ and with diameters tending to zero.

References

[1] G. Vitali, "Sui gruppi di punti e sulle funzioni di variabili reali" Atti Accad. Sci. Torino , 43 (1908) pp. 75–92 Zbl 39.0101.05
[2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05


Comments

For , Vitali's covering theorem is a main ingredient in the proof of the Lebesgue theorem that a monotone function has a finite derivative almost everywhere [a2].

There is another theorem that goes by the name Vitali convergence theorem. Let be a measure space, , a sequence in , and an -measurable function which is finite -almost-everywhere and such that -almost-everywhere. Then and if and only if: 1) for each there is a set such that and for all ; and 2) uniformly in . See [a2].

At least two other useful theorems bear Vitali's name. The Vitali theorem generalizing the Lebesgue's dominated convergence theorem for what is called an equi-integrable or uniformly integrable family of functions. There is also the Vitali–Hahn–Saks theorem, which asserts that a pointwise limit of a sequence of (-additive) measures on a -field is still a (-additive) measure.

References

[a1] H.L. Royden, "Real analysis", Macmillan (1968) pp. Chapt. 5
[a2] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[a3] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[a4] H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108 MR0257325 Zbl 0176.00801

Vitali's theorem on the uniform convergence of a sequence of holomorphic functions. Let a sequence of holomorphic functions on a domain of the complex -plane be uniformly bounded (cf. Uniform boundedness) and converge on a set with a limit point in ; the sequence will then converge uniformly inside towards a holomorphic function, i.e. will converge uniformly on every compact set . The theorem was obtained by G. Vitali .

The compactness principle makes it possible to strengthen Vitali's theorem by replacing the condition of uniform boundedness on by the condition of uniform boundedness on every compact set . There also exist Vitali theorems for normal families (cf. Normal family) of meromorphic functions, for families of quasi-analytic functions and for families of holomorphic functions of several complex variables; in the last case, however, additional limitations must be imposed on the set , for example, must contain interior points in [3], [4].

References

[1a] G. Vitali, Rend. R. Istor. Lombardo (2) , 36 (1903) pp. 772–774
[1b] G. Vitali, Ann. Mat. Pura Appl. (3) , 10 (1904) pp. 73
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) pp. Chapt.4 (Translated from Russian) MR0444912 Zbl 0357.30002
[3] P. Montel, "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars (1927) Zbl 53.0303.02
[4] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601

E.D. Solomentsev

Comments

References

[a1] C. Carathéodory, "Theory of functions of a complex variable" , 1 , Chelsea, reprint (1978) (Translated from German) MR1570711 MR0064861 MR0060009 Zbl 0056.06703 Zbl 0055.30301
[a2] J.B. Conway, "Functions of one complex variable" , Springer (1973) MR0447532 Zbl 0277.30001
[a3] R. Remmert, "Funktionentheorie" , II , Springer (1991) MR1150243 Zbl 0748.30002
How to Cite This Entry:
Vitali theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali_theorem&oldid=28282
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article