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Vitali theorem

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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
[2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)


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)
[a3] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[a4] H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108

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)
[3] P. Montel, "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars (1927)
[4] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)

E.D. Solomentsev

Comments

References

[a1] C. Carathéodory, "Theory of functions of a complex variable" , 1 , Chelsea, reprint (1978) (Translated from German)
[a2] J.B. Conway, "Functions of one complex variable" , Springer (1973)
[a3] R. Remmert, "Funktionentheorie" , II , Springer (1991)
How to Cite This Entry:
Vitali theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali_theorem&oldid=28282
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article