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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 $\mathbb R^n$.

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 [[#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.

====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  {{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====
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  $  (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 ( $  \sigma $-
additive) measures on a  $  \sigma $-
field is still a ( $  \sigma $-
additive) measure.


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]] 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]]].

''E.D. Solomentsev''

====References====
<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>
<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>
<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>