Vitali theorem
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 |
Vitali theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali_theorem&oldid=29167