Uniformly exhaustive submeasures and nearly additive set functions

Authors:
N. J. Kalton and James W. Roberts

Journal:
Trans. Amer. Math. Soc. **278** (1983), 803-816

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 one-dimensional subspace.

**[1]**Jens Peter Reus Christensen,*Some results with relation to the control measure problem*, Vector space measures and applications (Proc. Conf., Univ. Dublin, Dublin, 1977) Lecture Notes in Math., vol. 645, Springer, Berlin-New York, 1978, pp. 27–34. MR**502425****[2]**Wojchiech Herer and Jens Peter Reus Christensen,*On the existence of pathological submeasures and the construction of exotic topological groups*, Math. Ann.**213**(1975), 203–210. MR**0412369****[3]**Lech Drewnowski,*Un théorème sur les opérateurs de 𝑙_{∞}(Γ)*, C. R. Acad. Sci. Paris Sér. A-B**281**(1975), no. 22, Aii, A967–A969 (French, with English summary). MR**0385626****[4]**Ofer Gabber and Zvi Galil,*Explicit constructions of linear-sized superconcentrators*, J. Comput. System Sci.**22**(1981), no. 3, 407–420. Special issued dedicated to Michael Machtey. MR**633542**, 10.1016/0022-0000(81)90040-4**[5]**P. Hall,*On representatives of subsets*, J. London Math. Soc.**10**(1935), 26-30.**[6]**N. J. Kalton,*Linear operators whose domain is locally convex*, Proc. Edinburgh Math. Soc. (2)**20**(1976/77), no. 4, 293–299. MR**0454700****[7]**N. J. Kalton,*The three space problem for locally bounded 𝐹-spaces*, Compositio Math.**37**(1978), no. 3, 243–276. MR**511744****[8]**Gerald A. Goldin and Robert F. Wheeler (eds.),*Measure theory and its applications*, Northern Illinois University, Department of Mathematical Sciences, DeKalb, Ill., 1981. MR**613504****[9]**-,*Isomorphisms between spaces of vector-valued continuous functions*, Proc. Edinburgh Math. Soc. (to appear).**[10]**J. L. Kelley,*Measures on Boolean algebras*, Pacific J. Math.**9**(1959), 1165–1177. MR**0108570****[11]**Dorothy Maharam,*An algebraic characterization of measure algebras*, Ann. of Math. (2)**48**(1947), 154–167. MR**0018718****[12]**M. S. Pinsker,*On the complexity of a concentrator*, 318/1-318/4, 7th Internat. Telegraphic Conf., Stockholm, June 1973.**[13]**Nicholas Pippenger,*Superconcentrators*, SIAM J. Comput.**6**(1977), no. 2, 298–304. MR**0446750****[14]**V. A. Popov,*Additive and semiadditive functions on Boolean algebras*, Sibirsk. Mat. Ž.**17**(1976), no. 2, 331–339, 479 (Russian). MR**0419726****[15]**M. Ribe,*Examples for the nonlocally convex three space problem*, Proc. Amer. Math. Soc.**73**(1979), no. 3, 351–355. MR**518518**, 10.1090/S0002-9939-1979-0518518-9**[16]**J. W. Roberts,*Pathological compact convex sets in the spaces*, The Altgeld Book, University of Illinois, 1976.**[17]**James W. Roberts,*A compact convex set with no extreme points*, Studia Math.**60**(1977), no. 3, 255–266. MR**0470851****[18]**-,*A non-locally convex*-*space with the Hahn-Banach approximation property*, Banach Spaces of Analytic Functions, Lecture Notes in Math., vol. 604, Springer-Verlag, Berlin and New York, 1977, pp. 76-82.**[19]**S. Rolewicz and C. Ryll-Nardzewski,*On unconditional convergence in linear metric spaces*, Colloq. Math.**17**(1967), 327–331. MR**0219947****[20]**M. Simonard,*Linear programming*, Prentice-Hall, Englewood Cliffs, N. J., 1967.**[21]**Michel Talagrand,*A simple example of pathological submeasure*, Math. Ann.**252**(1979/80), no. 2, 97–102. MR**593624**, 10.1007/BF01420116

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1983-0701524-4

Keywords:
Submeasures,
control measure,
twisted sum

Article copyright:
© Copyright 1983
American Mathematical Society