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.
@article{bwmeta1.element.bwnjournal-article-smv112i2p165bwm,
author = {L. Rodr\'\i guez-Piazza},
title = {Derivability, variation and range of a vector measure},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {165-187},
zbl = {0824.46049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv112i2p165bwm}
}
Rodríguez-Piazza, L. Derivability, variation and range of a vector measure. Studia Mathematica, Tome 113 (1995) pp. 165-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv112i2p165bwm/
[00000] [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. | Zbl 0018.27601
[00001] [AD] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221-235. | Zbl 0749.28006
[00002] [BDS] R. G. Bartle, N. Dunford and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289-305. | Zbl 0068.09301
[00003] [B] E. D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323-346. | Zbl 0194.23102
[00004] [C] G. Choquet, Lectures on Analysis, Vol. III, W. A. Benjamin, Reading, Mass., 1969.
[00005] [DFJP] W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327. | Zbl 0306.46020
[00006] [D] J. Diestel, The Radon-Nikodym property and the coincidence of integral and nuclear operators, Rev. Roumaine Math. Pures Appl. 17 (1972), 1611-1620. | Zbl 0255.28010
[00007] [DU] J. Diestel and J. J. Uhl Jr., Vector Measures, Amer. Math. Soc., Providence, R.I., 1977.
[00008] [G] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
[00009] [KK] I. Kluvánek and G. Knowles, Vector Measures and Control Systems, North-Holland, Amsterdam, 1975.
[00010] [L] D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294-307. | Zbl 0242.28008
[00011] [M] G. Matheron, Un théorème d'unicité pour les hyperplans poissoniens, J. Appl. Probab. 11 (1974), 184-189. | Zbl 0287.60059
[00012] [N] A. Neyman, Decomposition of ranges of vector measures, Israel J. Math. 40 (1981), 54-64. | Zbl 0504.28013
[00013] [Pe] C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547. | Zbl 0103.15604
[00014] [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.
[00015] [R1] N. W. Rickert, Measures whose range is a ball, Pacific J. Math. 23 (1967), 361-371. | Zbl 0191.14303
[00016] [R2] N. W. Rickert, The range of a measure, Bull. Amer. Math. Soc. 73 (1967), 560-563. | Zbl 0153.38201
[00017] [Ri] M. A. Rieffel, The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968), 466-487. | Zbl 0169.46803
[00018] [Ro] L. Rodríguez-Piazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), 205-214. | Zbl 0727.28005
[00019] [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. | Zbl 0524.52002
[00020] [T] A. E. Tong, Nuclear mappings on C(X), Math. Ann. 194 (1971), 213-224.