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1994-1995 | 112 | 2 | 165-187
Tytuł artykułu

Derivability, variation and range of a vector measure

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove that the range of a vector measure determines the σ-finiteness of its variation and the derivability of the measure. Let F and G be two countably additive measures with values in a Banach space such that the closed convex hull of the range of F is a translate of the closed convex hull of the range of G; then F has a σ-finite variation if and only if G does, and F has a Bochner derivative with respect to its variation if and only if G does. This complements a result of [Ro] where we proved that the range of a measure determines its total variation. We also give a new proof of this fact.
Słowa kluczowe
Czasopismo
Rocznik
Tom
112
Numer
2
Strony
165-187
Opis fizyczny
Daty
wydano
1995
otrzymano
1993-10-21
poprawiono
1994-02-04
Twórcy
  • Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain, piazza@cica.es
Bibliografia
  • [A] A. D. Aleksandrov, Zur Theorie der gemischten Volumina von konvexen Körpern. II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen, Mat. Sb. (N.S.) 2 (1937), 1205-1238.
  • [AD] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221-235.
  • [BDS] R. G. Bartle, N. Dunford and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289-305.
  • [B] E. D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323-346.
  • [C] G. Choquet, Lectures on Analysis, Vol. III, W. A. Benjamin, Reading, Mass., 1969.
  • [DFJP] W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327.
  • [D] J. Diestel, The Radon-Nikodym property and the coincidence of integral and nuclear operators, Rev. Roumaine Math. Pures Appl. 17 (1972), 1611-1620.
  • [DU] J. Diestel and J. J. Uhl Jr., Vector Measures, Amer. Math. Soc., Providence, R.I., 1977.
  • [G] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
  • [KK] I. Kluvánek and G. Knowles, Vector Measures and Control Systems, North-Holland, Amsterdam, 1975.
  • [L] D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294-307.
  • [M] G. Matheron, Un théorème d'unicité pour les hyperplans poissoniens, J. Appl. Probab. 11 (1974), 184-189.
  • [N] A. Neyman, Decomposition of ranges of vector measures, Israel J. Math. 40 (1981), 54-64.
  • [Pe] C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547.
  • [P] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., Providence, R.I., 1986.
  • [R1] N. W. Rickert, Measures whose range is a ball, Pacific J. Math. 23 (1967), 361-371.
  • [R2] N. W. Rickert, The range of a measure, Bull. Amer. Math. Soc. 73 (1967), 560-563.
  • [Ri] M. A. Rieffel, The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968), 466-487.
  • [Ro] L. Rodríguez-Piazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), 205-214.
  • [SW] R. Schneider and W. Weil, Zonoids and related topics, in: Convexity and its Applications, P. M. Gruber and J. M. Wills (eds.), Birkhäuser, Basel (1983), 296-317.
  • [T] A. E. Tong, Nuclear mappings on C(X), Math. Ann. 194 (1971), 213-224.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv112i2p165bwm
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