Topological rings of sets and the theory of vector measures

Bogdan, V. M.; Oberle, R. A.

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Tytuł książki

Topological rings of sets and the theory of vector measures

Autorzy

V. M. Bogdan R. A. Oberle

Seria

Rozprawy Matematyczne tom/nr w serii: 154 wydano: 1978

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CONTENTS

Introduction............................................................................................................................................................... 5

Chapter I. Topological rings of sets
1.1. Definition and basic properties of topological rings of sets............................................................... 7
1.2. Topological rings of sets generated by Rickart families of contents................................................ 12

Chapter II. The space of Rickart vector charges on a ring of sets
2.1. Definition and basic properties of Rickart vector charges................................................................... 23
2.2. Pointwise convergent sequences of Rickart vector charges.............................................................. 28

Chapter III. Equicontinuous sequences of Rickart vector charges
3.1. Vectorial generalizations of the Nikodym boundedness theorem..................................................... 35
3.2. Generalizations of the Vitali-Hahn-Saks theorem................................................................................. 39

Chapter IV. Weak compactness and decompositions of strongly bounded vector charges
4.1. Weak compactness in the spaces of finitely additive scalar charges on
an algebra of sets................................................................................................................................................ 42
4.2. Decompositions of strongly bounded vector charges.......................................................................... 45

Chapter V. Extensions of Rickart vector measures
5.1. Extensions of countably additive Rickart vector measures.................................................................. 52
5.2. General extension of vector measures.................................................................................................... 55

Summary of definitions............................................................................................................................................ 67

References................................................................................................................................................................. 68

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 154

Liczba stron

70

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CLIV

Daty

wydano

1978

Twórcy

autor

V. M. Bogdan

autor

R. A. Oberle

Bibliografia

[1] T. Ando, Convergent sequences of finitely additive measures, Pacific J. Math. 11 (1960), pp. 395-404.
[2] G. Ya. Areskin, On the compactness of the family of completely additive set functions, (Russian), Leningrad Gos. Pod. Inst. Ucen. Zap. 238 (1962), pp. 102-118.
[3] E. G. Bartle, A general bilinear vector integral, Studia Math. 15 (1956), pp. 337-352.
[4] R. G. Bartle, N. Dunford and J. T. Schwartz, Weak compactness and vector measures, Canadian J. Math. 7 (1956), pp. 287-305.
[5] K. Bichteler, Integration theory, Springer Verlag, New York 1973.
[6] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series on Banach spaces, Studia Math. 17 (1958), pp. 151-164.
[7] S. Bochner and R. S. Phillips, Additive set functions and vector lattices, Annals Math. 42 (1941), pp. 316-324.
[8] W. M. Bogdanowicz, A generalization of the Lebesgue-Bochner-Stieltjes integral and a new approach to the theory of integration, Proceedings of the National Academy of Sciences (USA) 53 (3) (1956), pp. 492-498.
[9] W. M. Bogdanowicz, Existence and uniqueness of extensions of volumes and the operation of completion of a volume I, Proceedings of the Japan Academy 42 (6) (1966), pp. 571-575.
[10] W. M. Bogdanowicz, An approach to the theory of Lebesgue-Bochner measurable functions and to the theory of measure, Math. Annalen 164 (1966), pp. 215-270.
[11] W. M. Bogdanowicz, Relations between the Lebesgue integral generated by a measure and the integral generated by a volume, Annales Societatis Mathematicae Polonae, Seria I, Commentationes Mathematicae 12 (1969), pp. 277-299.
[12] J. K. Brooks, On the existence of a control measure for strongly bounded vector measures, Bulletin Amer. Math. Soc. 77 (6) (1972), pp. 999-1001.
[13] J. K. Brooks, Weak compactness in the space of vector measures, ibid. 78 (2) (1972), pp. 284-287.
[14] J. K. Brooks, On the existence of a control measure for strongly bounded vector measures, Notices Amer. Math. Soc. 18 (2) (1971), p. 45.
[15] J. K. Brooks and R. S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci. (USA) 67 (3) (1970), pp. 1294-1298.
[16] R. B. Darst, A decomposition of finitely additive set functions, J. Math. Angew. 210 (1962), pp. 31-37.
[17] R. B. Darst, A direct proof of Porcelli's weak compactness criteria. Proc. Amer. Math. Soc. 17 (1966), pp. 1094-1096.
[18] R. B. Darst, On a theorem of Nikodym with applications to Von Neumann Algebras, Bull. Amer. Math. Soc. 74 (2) (1968), pp. 283-284.
[19] J. Diestel, Abstract valued additive set functions of locally finite variations, Notices Amer. Math. Soc. 17 (1970), p. 657.
[20] N. Dinculeanu, Vector measures, Pergamon Press, New York 1967.
[21] L. Drewnowski, Topological rings of sets, continuous set functions, integration, I, II, and III, Bull. Acad. Polon. Sci., Sér. Sci. math. astr. et phys. 20 (4) (1972), pp. 269-276, 277-286, and 439-445.
[22] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience Publishers, Inc., New York 1966.
[23] G. Fox, Inductive extension of a vector measure under a convergence condition, Canad. J. Math. 20 (1968), pp. 1246-1255.
[24] G. G. Gould, Extensions of vector valued measures, Proc. London Math. Soc. 16 (3) (1966), pp. 685-704.
[25] L. J. Heider, A representation theory for measures on Boolean algebras, Michigan Math. J. 5 (1958), pp. 213-221.
[26] E. Huff, The Yosida-Hewitt decomposition as an ergodic theorem. Vector and operator valued measures, Academic Press 1974, pp. 133-139.
[27] I. Kluvanek, The extension and closure of a rector measure. Vector and operator valued measures, Academic Press, New York, 1974, pp. 175-190.
[28] I. Labuda, Sur quelques generalizations des theorems de Nikodym el de Vitali-Hahn-Saks, Bull. Acad. Polon. Sci., Sér. sci. math. astr. et phys. 20 (6) (1972), pp. 447-456.
[29] S. P. Lloyd, On finitely additive set functions, Proc. Amer. Math. Soc. 14 (1963), pp. 701-704.
[30] C. W. Mc Arthur, On a theorem of Orlicz-Pettis, Pacific J. Math. 22 (1967), pp. 292-302.
[31] M. Nakamura and G. Sunouchi, Note on Banach spaces (IV): On a decomposition of additive set functions, Proceedings of the Imperial Academy XVIII (1942), pp. 333-335.
[32] H. Nakano, Linear lattices, Wayne State University Press, Detroit 1966.
[33] O. Nikodym, Sur les suites convergentes dc fonctiones parfaitemente additives d'ensemble abstract, Monatsh. für Math. u. Phys. 40 (1933), pp. 427-432.
[34] R. Oberle, Theory of a class of vector measures on topological rings of sets and generalizations of the Vitali-Hahn-Salcs Theorem, (dissertation) The Catholic Univ. America, 1971.
[35] S. Ohba, The decomposition theorems for vector measures. The Yokohama Math. J. XIX (1) (1971), pp. 23-28.
[36] S. Ohba, Decompositions of vector measures. The Reports of the Faculty of Technology, Kanagaw Univ., No. 10, 10, 1972.
[37] A. L. Peressini, Ordered topological vector spaces, Harper and Row, Publishers, Now York 1967.
[38] R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), pp. 516-541.
[39] P. Porcelli, Two embedding theorems with applications to weak convergence and compactness in spaces of additive type functions, J. Math. Mech. 9#2, (1960), pp. 273-292.
[40] C. E. Rickart, Decomposition of additive set functions, Duke Math. J. 10 (1943), pp. 653-665.
[41] A. P. Robertson, On unconditional convergence in topological vector spaces, Proc. Royal Society Edinburgh, Section A68 (1969), pp. 145-147.
[42] S. Saks, On some functionals I, Trans. Amer. Math. Soc. 35 (1933), pp. 549-556.
[43] S. Saks, Additional to the note on some functionals, ibid. 35 (1933), pp. 967-974.
[44] M. Sion, Outer measures with values in a topological group, Proc. London Math. Soc. 19 (3) (1969), pp. 89-106.
[45] T. Traynor, Decompositions of group valued additive set functions, Ann. de l'Inst. Fourier, XXII (1972).
[46] T. Traynor, A general Hewitt-Yosida decomposition, To appear, Canadian Journal of Mathematics.
[47] T. Traynor, S-bounded additive set functions, in: Vector and operator valued measures, Academic Press, 1974, pp. 355-365.
[48] D. H. Tucker and H. Maynard, Vector and operator valued measures, Academic Press, New York 1974.
[49] J. J. Uhl, Extensions and decompositions of vector measures, J. London Math. Soc. 3 (1971), pp. 672-676.

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Topological rings of sets and the theory of vector measures

Autorzy

Seria

Zawartość

Warianty tytułu

Abstrakty

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Copyright

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