Vitali theorem

Vitali's covering theorem. If a system of closed sets is a Vitali covering (see below) of a set , it is possible to extract from an at most countable sequence of pairwise disjoint sets , such that

where is the outer Lebesgue measure in .

A Vitali covering of a set is a system of subsets of such that for any there exists a sequence from satisfying the following conditions:

(1)

(2)

where is the diameter of ; and

(3)

where the supremum is taken over all (cubes with faces parallel to the coordinate planes and containing ), and where is the outer Lebesgue measure in ; this supremum is said to be the regularity parameter of .

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

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