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====References====
 
====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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>
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<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>
  
  
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====References====
 
====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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer  (1969)  pp. 60; 62; 71; 108</TD></TR></table>
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<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></table>
  
 
Vitali's theorem on the uniform convergence of a sequence of holomorphic functions. Let a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678055.png" /> of holomorphic functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678056.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678057.png" />-plane be uniformly bounded (cf. [[Uniform boundedness|Uniform boundedness]]) and converge on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678058.png" /> with a limit point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678059.png" />; the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678060.png" /> will then converge uniformly inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678061.png" /> towards a holomorphic function, i.e. will converge uniformly on every compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678062.png" />. The theorem was obtained by G. Vitali .
 
Vitali's theorem on the uniform convergence of a sequence of holomorphic functions. Let a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678055.png" /> of holomorphic functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678056.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678057.png" />-plane be uniformly bounded (cf. [[Uniform boundedness|Uniform boundedness]]) and converge on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678058.png" /> with a limit point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678059.png" />; the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678060.png" /> will then converge uniformly inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678061.png" /> towards a holomorphic function, i.e. will converge uniformly on every compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096780/v09678062.png" />. The theorem was obtained by G. Vitali .
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====References====
 
====References====
<table><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)</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)</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)</TD></TR></table>
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<table><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></table>
  
 
''E.D. Solomentsev''
 
''E.D. Solomentsev''
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====References====
 
====References====
<table><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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Remmert,  "Funktionentheorie" , '''II''' , Springer  (1991)</TD></TR></table>
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<table><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>

Revision as of 12:13, 27 September 2012

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

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
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