Uniformly exhaustive submeasures and nearly additive set functions
Authors:
N. J. Kalton and James W. Roberts
Journal:
Trans. Amer. Math. Soc. 278 (1983), 803816
MSC:
Primary 28A60; Secondary 46A06
MathSciNet review:
701524
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Abstract: Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an space has a control measure. We also show that (or any space) is a space, i.e. cannot be realized as the quotient of a nonlocally convex space by a onedimensional subspace.
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 [1]
 J. P. R. Christensen, Some results with relation to the control measure problem, Vector Space Measures and Applications. II, Lecture Notes in Math., vol. 645, SpringerVerlag, Berlin and New York, 1978, pp. 2734. MR 502425 (80e:28008)
 [2]
 J. P. R. Christensen and W. Herer, On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann. 213 (1975), 203210. MR 0412369 (54:495)
 [3]
 L. Drewnowski, Un théorème sur les opérateurs de , C. R. Acad. Sci.Paris Sér. A 281 (1976), 967969. MR 0385626 (52:6486)
 [4]
 O. Gabber and Z. Galil, Explicit constructions of linear size superconcentrators, J. Comput. System Sci. 22 (3) (1981), 407420. MR 633542 (83d:94029)
 [5]
 P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 2630.
 [6]
 N. J. Kalton, Linear operators whose domain is locally convex, Proc. Edinburgh Math. Soc. 20 (1976), 293299. MR 0454700 (56:12949)
 [7]
 , The three space problem for locally bounded spaces, Compositio Math. 37 (1978), 243276. MR 511744 (80j:46005)
 [8]
 , Problem 5 (p. 284), Measure Theory and Its Applications (Proc. Conf., Northern Illinois Univ., 1980, G. A. Goldin and R. F. Wheeler, editors), DeKalb, Illinois, 1981. MR 613504 (82g:28002)
 [9]
 , Isomorphisms between spaces of vectorvalued continuous functions, Proc. Edinburgh Math. Soc. (to appear).
 [10]
 J. L. Kelley, Measures on Boolean algebras, Pacific J. Math. 9 (1959), 11651177. MR 0108570 (21:7286)
 [11]
 D. Maharam, An algebraic characterization of measure algebras, Ann. of Math. (2) 48 (1947), 154167. MR 0018718 (8:321b)
 [12]
 M. S. Pinsker, On the complexity of a concentrator, 318/1318/4, 7th Internat. Telegraphic Conf., Stockholm, June 1973.
 [13]
 N. Pippenger, Superconcentrators, SIAM J. Comput. 6 (1977), 298304. MR 0446750 (56:5074)
 [14]
 V. A. Popov, Additive and subadditive functions on Boolean algebras, Siberian Math. J. 17 (1976), 331339. (Russian) MR 0419726 (54:7744)
 [15]
 M. Ribe, Examples for the nonlocally convex threespace problem, Proc. Amer. Math. Soc. 73 (1979), 351355. MR 518518 (81a:46010)
 [16]
 J. W. Roberts, Pathological compact convex sets in the spaces , The Altgeld Book, University of Illinois, 1976.
 [17]
 , A compact convex set with no extreme points, Studia Math. 60 (1977), 255266. MR 0470851 (57:10595)
 [18]
 , A nonlocally convex space with the HahnBanach approximation property, Banach Spaces of Analytic Functions, Lecture Notes in Math., vol. 604, SpringerVerlag, Berlin and New York, 1977, pp. 7682.
 [19]
 S. Rolewicz and C. RyllNardzewski, On unconditional convergence in linear metric spaces, Colloq. Math. 17 (1967), 327331. MR 0219947 (36:3017)
 [20]
 M. Simonard, Linear programming, PrenticeHall, Englewood Cliffs, N. J., 1967.
 [21]
 M. Talagrand, A simple example of a pathological submeasure. Math. Ann. 252 (1980), 97102. MR 593624 (81k:28005)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198307015244
PII:
S 00029947(1983)07015244
Keywords:
Submeasures,
control measure,
twisted sum
Article copyright:
© Copyright 1983
American Mathematical Society
