Proceedings of the American Mathematical Society

Author(s): Peter G. Ovchinnikov
Journal: Proc. Amer. Math. Soc. 127 (1999), 1957-1966.
MSC (1991): Primary 06C15; Secondary 81P10
Posted: February 26, 1999
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Abstract: We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets.

1.: L. Beran, Orthomodular lattices. Algebraic approach, Academia, Prague, 1984. MR 86m:06015a
2.: G. Birkhoff, Lattice theory, AMS, Providence, RI, 1967. MR 37:2638
3.: G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. of Math. 37 (1936), 823-843.
4.: L. Bunce and J. Maitland Wright, The Mackey-Gleason problem, Bull. Amer. Math. Soc. 26 (1992), 288-293. MR 92i:46070
5.: E. Christensen, Measures on projections and physical states, Comm. Math. Phys. 86 (1982), 529-538. MR 85b:46072
6.: P. Cohn, Universal algebra, Harper & Row, New York, 1965. MR 31:224
7.: J. Dravecký and J. \v{S}ipo\v{s}, On the additivity of Gudder integral, Math. Slovaca 30 (1980), 299-303. MR 82b:28009
8.: A. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. MR 20:2609
9.: S. Gudder, Stochastic methods in quantum mechanics, North Holland, New York, 1979.
10.: -, An extension of classical measure theory, SIAM Rev. 26 (1984), 71-89. MR 85d:81014
11.: S. Gudder and J.-P. Marchand, A coarse-grained measure theory, Bull. Acad. Pol. Sc. 28 (1980), 557-564. MR 82i:28004
12.: R. Horn and C. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1986. MR 87e:15001
13.: G. Kalmbach, Orthomodular lattices, Academic Press, London, 1983. MR 85f:06012
14.: M. Matveichuk, A theorem on states on quantum logics, Teor. Mat. Fizika 45 (2) (1980), 244-250 (Russian). MR 82d:81015
15.: -, A theorem on states on quantum logics, II, Teor. Mat. Fizika 48 (3) (1981), 261-265 (Russian). MR 83a:81004
16.: -, The description of finite measures in semifinite algebras, Funk. Anal. Prilozhen. 15 (3) (1981), 41-53 (Russian).
17.: R. Mayet and M. Navara, Classes of logics representable as kernels of measures, in Contr. General Algebra 9 (G.Pilz, ed.), Teubner, Stuttgart/Wien, 1995, pp. 241-248.
18.: M. Navara, When is the integral on quantum probability spaces additive?, Real Anal. Exchange 14 (1989), 228-234. MR 90c:81018
19.: -, Kernel logics, Tatra Mountains Math. Publ. 3 (1993), 27-30. MR 95e:06024
20.: -, Quantum logics representable as kernels of measures, Czechoslovak Math. J. 46 (1996), 587-597. MR 97j:81027
21.: M. Navara and P. Pták, Two-valued measures on $\sigma$ -classes, \v{C}as. P\v{e}st. Mat. 108 (1983), 225-229. MR 84m:28006
22.: P. Ovchinnikov, Structure of measures on quantum logics, Thesis, Kazan State University, Kazan, Russia, 1985 (Russian).
23.: -, Problem 4, in Proc. Second Winter School on Measure Theory, Liptovský Ján, January 7-12, 1990, Czecho-Slovakia (A. Dvure\v{c}enskij and S. Pulmannová, eds.), p. 219.
24.: -, Measures on the logics of Gudder and Marchand, Konstr. Teoriya Funktsi[??]i i Funk. Analiz (Kazan) 8 (1992), 95-98 (Russian). MR 94e:81020
25.: -, A Galois correspondence connected with extending signed measures on a $\sigma$ -class of subsets of a finite set, Izv. Vyssh. Uchebn. Zaved. Mat. (5) (1994), 36-40 (Russian). MR 95i:81019
26.: R. Prather, Generating the $k$ -subsets of an $n$ -set, Amer. Math. Monthly 87 (1980), 740-743.
27.: P. Pták and S. Pulmannová, Orthomodular structures as quantum logics, Kluwer, Dordrecht, 1991. MR 82e:05005
28.: A. Sherstnev, On Boolean logics, Uch. Zap. KGU 128 (1968), 48-62 (Russian).
29.: F. Sultanbekov, Signed measures and automorphisms of a class of finite concrete logics, Konstr. Teoriya Funktsi[??]i i Funk. Analiz (Kazan) 8 (1992), 57-68 (Russian). MR 94f:81010
30.: -, Set logics and their representations, Int. J. Theor. Phys. 32 (1993), 2177-2186. MR 95c:06019
31.: F. Yeadon, Measures on projections in $W^{*}$ -algebras of type $II_{1}$ , Bull. London Math. Soc. 15 (1983), 139-145. MR 84g:46089
32.: -, Finitely additive measures on projections in finite $W^{*}$ -algebras, Bull. London Math. Soc. 16 (1984), 145-150. MR 85i:46087
33.: J. Zerbe and S. Gudder, Additivity of integrals on generalized measure spaces, J. Comb. Theory (A) 39 (1985), 42-52. MR 86m:81020

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 06C15, 81P10

Peter G. Ovchinnikov
Affiliation: Department of Mathematics, Kazan State University, 420008, Kazan, Russia
Email: Petr.Ovchinnikov@ksu.ru

DOI: 10.1090/S0002-9939-99-04761-9
PII: S 0002-9939(99)04761-9
Keywords: Finite concrete logic, measure, signed measure
Received by editor(s): August 15, 1996
Received by editor(s) in revised form: October 8, 1997
Posted: February 26, 1999
Additional Notes: This research was supported by the Russian Foundation for Fundamental Research, grants no. 95--01--00025 and no. 96--01--01265.
Communicated by: Andreas R. Blass
Copyright of article: Copyright 1999, American Mathematical Society

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