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In [4] it is proved that a measure on a finite coarse-grained space extends, as a signed measure, over the entire power algebra. In [7] this result is reproved and further improved. Both the articles [4] and [7] use the proof techniques of linear spaces (i.e. they use multiplication by real scalars). In this note we show that all the results cited above can be relatively easily obtained by the Horn-Tarski extension technique in a purely combinatorial manner. We also characterize the pure measures and settle the dimension of the normalized-measure space. We then comment on a consequence of the results for circulant matrices. Finally, we take up the case of circle coarse-grained space and also establish a measure-extension result.
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Rocznik
Tom
Strony
1--11
Opis fizyczny
Bibliogr. 9 poz.
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autor
autor
- Department of Mathematics and Statistics, University Federico II, Naples Complesso Monte S. Angelo, via Cintia 80126 Naples, Italy, annades@unina.it
Bibliografia
- [1] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges, Academic Press,1983.
- [2| P. J. Davis, Circulant Matrices, Wiley, New York, 1979.
- [3] S. Gudder, An extension of classical measure theory, SIAM Rev. 26 (1984), 71-89.
- [4] S. Gudder and J. P. Marchand, A coarse-grained measure theory, Bull. Polish Acad. Sci. Math. 28 (1980), 557-564.
- [5] A. Horn and A. Tarski, Measures in Boolean algebras, Trans. Amer. Math. Soc. 64 (1948), 467-497.
- [6] M. Navara and P. Ptak, Almost Boolean orthomodular posets, J. Pure Appl. Algebra 60 (1989), 105-111.
- [7] P. G. Ovchinnikov, Measures on the Gudder-Marchand logics, in: Constructive Theory of Functions and Functional Analysis 8, Kazan. Gos. Univ., 1992, 95-98.
- [8] R. E. Prather, Generating the k-subsets of an n-set, Amer. Math. Monthly 87 (1980), 740-743.
- [9] P. Ptak and S. Pulmannova, Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht, 1991
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Bibliografia
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bwmeta1.element.baztech-article-BAT5-0011-0010