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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Families of sets of positive measure


Author: Grzegorz Plebanek
Journal: Trans. Amer. Math. Soc. 332 (1992), 181-191
MSC: Primary 28A12; Secondary 28C15
DOI: https://doi.org/10.1090/S0002-9947-1992-1044965-6
MathSciNet review: 1044965
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Abstract: We present a combinatorial description of those families $ \mathcal{P}$ of sets, for which there is a finite measure $ \mu $ such that $ \inf \{ \mu (P):P \in \mathcal{P}\} > 0$. This result yields a topological characterization of measure-compactness and Borel measure-compactness. It is also applied to a problem on the existence of regular measure extensions.


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  • [1] W. Adamski, Extensions of tight set functions with applications in topological measure theory, Trans. Amer. Math. Soc. 283 (1984), 353-368. MR 735428 (86c:28007)
  • [2] G. Bachmann and R. Cohen, Regular lattice measures and repleteness, Comm. Pure Appl. Math. 26 (1973), 587-599. MR 0335750 (49:530)
  • [3] G. Bachmann and A. Sultan, On regular extensions of measures, Pacific J. Math. 86 (1980), 389-395. MR 590550 (82f:28016)
  • [4] K. P. S. Bhaskara Rao and B. V. Rao, Theory of charges, Academic Press, New York, 1983. MR 751777 (86f:28006)
  • [5] J. A. Van Casteren, Supports of Borel measures, Univ. Antwerpen, 1974, preprint.
  • [6] W. W. Comfort and S. Negrepontis, Chain conditions in topology, Cambridge Univ. Press, 1982. MR 665100 (84k:04002)
  • [7] K. P. Dalgas, A general extension theorem for group-valued measures, Math. Nachr. 106 (1982), 153-170. MR 675753 (84f:28005)
  • [8] R. J. Gardner and W. F. Pfeffer, Borel measures, Handbook of Set-Theoretic Topology, Elsevier, 1984. MR 776641 (86c:28031)
  • [9] P. R. Halmos, Measure theory, Van Nostrand, 1950. MR 0033869 (11:504d)
  • [10] W. Josephson, Coallocation between lattices with applications to measure extensions, Pacific J. Math. 75 (1978), 149-163. MR 0492154 (58:11303)
  • [11] J. L. Kelley, Measures on Boolean algebras, Pacific J. Math. 9 (1959), 1165-1177. MR 0108570 (21:7286)
  • [12] Z. Lipecki, Tight extensions of group-valued quasi-measures, Colloq. Math. 51 (1987), 213-219. MR 891289 (88m:28005)
  • [13] J. Lembcke, Konservative Abbildungen und Fortsetzung regulärer Masse, Z. Wahrsch. Verw. Gebiete 15 (1970), 57-96. MR 0276429 (43:2176)
  • [14] G. Mägerl and I. Namioka, Intersection numbers and $ wea{k^{\ast} }$ separability of spaces and measures, Math. Ann. 249 (1980), 273-279. MR 579106 (81j:28022)
  • [15] E. Marczewski (Szpilrajn), Remarques sur les fonctions completement additives et sur les ensembles jouissant de la propriete de Baire, Fund. Math. 22 (1937), 303-311.
  • [16] E. Marczewski, On compact measures, Fund. Math. 40 (1953), 113-124. MR 0059994 (15:610a)
  • [17] J. Pachl, Disintegration and compact measures, Math. Scand. 43 (1978), 157-168. MR 523833 (80d:28020)
  • [18] G. Plebanek, On strictly positive measures on topological spaces, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 181-191. MR 1111767 (92f:28011)
  • [19] R. F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 2 (1983), 9-190. MR 710569 (85b:46035)
  • [20] M. Wilhelm, Existence of additive functional on semigroups and the von Neumann minimax theorem, Colloq. Math. 35 (1976), 268-274. MR 0414133 (54:2237)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1044965-6
Article copyright: © Copyright 1992 American Mathematical Society

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