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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
+
{{TEX|done}}
  
<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>
+
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 $\mathbb R^n$.
  
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" />.
+
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 $E_n$.
  
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:
+
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.
 
 
<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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Vitali,  "Sui gruppi di punti e sulle funzioni di variabili reali"  ''Atti Accad. Sci. Torino'' , '''43'''  (1908)  pp. 75–92</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Vitali,  "Sui gruppi di punti e sulle funzioni di variabili reali"  ''Atti Accad. Sci. Torino'' , '''43'''  (1908)  pp. 75–92 {{MR|}}  {{ZBL|39.0101.05}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French) {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}}  {{ZBL|63.0183.05}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678033.png" />, 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 [[#References|[a2]]].
+
For $  n =1 $,  
 +
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 [[#References|[a2]]].
  
There is another theorem that goes by the name Vitali convergence theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678034.png" /> be a [[Measure space|measure space]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678036.png" /> a sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678037.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678038.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678039.png" />-measurable function which is finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678040.png" />-almost-everywhere and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678041.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678042.png" />-almost-everywhere. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678044.png" /> if and only if: 1) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678045.png" /> there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678046.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678048.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678049.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678050.png" /> uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678051.png" />. See [[#References|[a2]]].
+
There is another theorem that goes by the name Vitali convergence theorem. Let $  (X,\  {\mathcal A} ,\  \mu ) $
 +
be a [[Measure space|measure space]], $  1 \leq p < \infty $,  
 +
$  (f _{n} ) _{n=1} ^ \infty  $
 +
a sequence in $  L _{p} (X) $,  
 +
and $  f $
 +
an $  {\mathcal A} $-
 +
measurable function which is finite $  \mu $-
 +
almost-everywhere and such that $  f _{n} \rightarrow f $
 +
$  \mu $-
 +
almost-everywhere. Then $  f \in L _{p} (X) $
 +
and $  \| f - f _{n} \| _{p} \rightarrow 0 $
 +
if and only if: 1) for each $  \epsilon > 0 $
 +
there is a set $  A _ \epsilon  \in {\mathcal A} $
 +
such that $  \mu (A _ \epsilon  ) < \infty $
 +
and $  \int _{ X \setminus A _ \epsilon} | f _{n} | ^{p} \  d \mu < \epsilon $
 +
for all $  n \in \mathbf N $;  
 +
and 2) $  \lim\limits _{ {\mu (E) \rightarrow 0}} \  \int _{E} | f _{n} | ^{p} \  d \mu = 0 $
 +
uniformly in $  n $.  
 +
See [[#References|[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 (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678052.png" />-additive) measures on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678053.png" />-field is still a (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678054.png" />-additive) measure.
+
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 ( $  \sigma $-
 +
additive) measures on a $  \sigma $-
 +
field is still a ( $  \sigma $-
 +
additive) measure.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.L. Royden,   "Real analysis" , Macmillan  (1968)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer  (1969)  pp. 60; 62; 71; 108</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.L. Royden, [[Royden, "Real analysis"|"Real analysis"]], Macmillan  (1968)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958) {{MR|0117523}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer  (1969)  pp. 60; 62; 71; 108 {{MR|0257325}} {{ZBL|0176.00801}} </TD></TR></table>
  
Vitali's theorem on the uniform convergence of a sequence of holomorphic functions. Let a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678055.png" /> of holomorphic functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678056.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678057.png" />-plane be uniformly bounded (cf. [[Uniform boundedness|Uniform boundedness]]) and converge on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678058.png" /> with a limit point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678059.png" />; the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678060.png" /> will then converge uniformly inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678061.png" /> towards a holomorphic function, i.e. will converge uniformly on every compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678062.png" />. The theorem was obtained by G. Vitali .
+
Vitali's theorem on the uniform convergence of a sequence of holomorphic functions. Let a sequence $  \{ f _{n} (z) \} $
 +
of holomorphic functions on a domain $  D $
 +
of the complex $  z $-
 +
plane be uniformly bounded (cf. [[Uniform boundedness|Uniform boundedness]]) and converge on a set $  E $
 +
with a limit point in $  D $;  
 +
the sequence $  \{ f _{n} (z) \} $
 +
will then converge uniformly inside $  D $
 +
towards a holomorphic function, i.e. will converge uniformly on every compact set $  K \subset D $.  
 +
The theorem was obtained by G. Vitali .
  
The [[Compactness principle|compactness principle]] makes it possible to strengthen Vitali's theorem by replacing the condition of uniform boundedness on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678063.png" /> by the condition of uniform boundedness on every compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678064.png" />. There also exist Vitali theorems for normal families (cf. [[Normal family|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678065.png" />, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678066.png" /> must contain interior points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678067.png" /> [[#References|[3]]], [[#References|[4]]].
+
The [[Compactness principle|compactness principle]] makes it possible to strengthen Vitali's theorem by replacing the condition of uniform boundedness on $  D $
 +
by the condition of uniform boundedness on every compact set $  K \subset D $.  
 +
There also exist Vitali theorems for normal families (cf. [[Normal family|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 $  E \subset D \subset \mathbf C ^{n} $,  
 +
for example, $  E $
 +
must contain interior points in $  \mathbf C ^{n} $[[#References|[3]]], [[#References|[4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  G. Vitali,  ''Rend. R. Istor. Lombardo (2)'' , '''36'''  (1903)  pp. 772–774</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  G. Vitali,  ''Ann. Mat. Pura Appl. (3)'' , '''10'''  (1904)  pp. 73</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  pp. Chapt.4  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Montel,  "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars  (1927)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>
+
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  G. Vitali,  ''Rend. R. Istor. Lombardo (2)'' , '''36'''  (1903)  pp. 772–774</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  G. Vitali,  ''Ann. Mat. Pura Appl. (3)'' , '''10'''  (1904)  pp. 73</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  pp. Chapt.4  (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Montel,  "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars  (1927) {{MR|}}  {{ZBL|53.0303.02}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR></table>
  
 
''E.D. Solomentsev''
 
''E.D. Solomentsev''
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Carathéodory,  "Theory of functions of a complex variable" , '''1''' , Chelsea, reprint  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Remmert,  "Funktionentheorie" , '''II''' , Springer  (1991)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Carathéodory,  "Theory of functions of a complex variable" , '''1''' , Chelsea, reprint  (1978)  (Translated from German) {{MR|1570711}} {{MR|0064861}} {{MR|0060009}} {{ZBL|0056.06703}} {{ZBL|0055.30301}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1973) {{MR|0447532}} {{ZBL|0277.30001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Remmert,  "Funktionentheorie" , '''II''' , Springer  (1991) {{MR|1150243}} {{ZBL|0748.30002}} </TD></TR></table>

Latest revision as of 09:05, 19 September 2021


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 $ n =1 $, 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 $ (X,\ {\mathcal A} ,\ \mu ) $ be a measure space, $ 1 \leq p < \infty $, $ (f _{n} ) _{n=1} ^ \infty $ a sequence in $ L _{p} (X) $, and $ f $ an $ {\mathcal A} $- measurable function which is finite $ \mu $- almost-everywhere and such that $ f _{n} \rightarrow f $ $ \mu $- almost-everywhere. Then $ f \in L _{p} (X) $ and $ \| f - f _{n} \| _{p} \rightarrow 0 $ if and only if: 1) for each $ \epsilon > 0 $ there is a set $ A _ \epsilon \in {\mathcal A} $ such that $ \mu (A _ \epsilon ) < \infty $ and $ \int _{ X \setminus A _ \epsilon} | f _{n} | ^{p} \ d \mu < \epsilon $ for all $ n \in \mathbf N $; and 2) $ \lim\limits _{ {\mu (E) \rightarrow 0}} \ \int _{E} | f _{n} | ^{p} \ d \mu = 0 $ uniformly in $ n $. 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 ( $ \sigma $- additive) measures on a $ \sigma $- field is still a ( $ \sigma $- 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 $ \{ f _{n} (z) \} $ of holomorphic functions on a domain $ D $ of the complex $ z $- plane be uniformly bounded (cf. Uniform boundedness) and converge on a set $ E $ with a limit point in $ D $; the sequence $ \{ f _{n} (z) \} $ will then converge uniformly inside $ D $ towards a holomorphic function, i.e. will converge uniformly on every compact set $ K \subset D $. 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 $ D $ by the condition of uniform boundedness on every compact set $ K \subset D $. 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 $ E \subset D \subset \mathbf C ^{n} $, for example, $ E $ must contain interior points in $ \mathbf C ^{n} $[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=17087
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article