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 $n =1$, 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 $\int _{ {A _ \epsilon}} | f _{n} | ^{p} \ d \mu < \epsilon$ 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

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