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Volume 68, Issue 3


Approximation in quantum measure spaces

Mona Khare / Bhawna Singh / Anurag Shukla
Published Online: 2018-05-18 | DOI: https://doi.org/10.1515/ms-2017-0119


The objective of the present paper is to introduced and study the notion of μ-approximation by a subfamily K of a difference poset E. Various properties are proved and then applied to obtain some crucial results including a generalization of the Marczewski Theorem which states that countable compactness is sufficient for σ-additivity of a supermodular measure μ.

MSC 2010: Primary 06C15, 81P10, 03G10; Secondary 28C99

Keywords: difference posets; σ-additive measure; compact measure; μ-approximation

The third author acknowledges with gratitude the financial support in part by Council of Scientific and Industrial Research (CSIR), New Delhi, India under Grant No. 09/001(0320) /2009-EMR-I.


  • [1]

    Avallone, A.—Basile, A.: On a Marinacci uniqueness theorem for measures, J. Math. Anal. Appl. 286 (2003), 378–390.CrossrefGoogle Scholar

  • [2]

    Beltrametti, E. G.—Cassinelli, G.: The Logic of Quantum Mechanics, Addison-Wesley Publishing Co., Reading, Massachusetts, 1981.Google Scholar

  • [3]

    Bennet, M. K.—Foulis, D. J.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331–1352.CrossrefGoogle Scholar

  • [4]

    Beran, L.: Orthomodular Lattices, Algebraic Approach, D. Reidel, Holland, 1984.Google Scholar

  • [5]

    Birkhoff, G.—Von Neumann, J.: The logic of quantum mechanics, Ann. Math. 37 (1936), 823–834.CrossrefGoogle Scholar

  • [6]

    Butnariu, D.—Klement, P., Triangular Norm-based Measures and Games with Fuzzy Coalitions, Kluwer Acad. Pub., Dordrecht, 1993.Google Scholar

  • [7]

    Dvurečenskij, A.—Pulmannová, S.: Difference posets, effects and quantum measurements, Internat. J. Theoret. Phys. 33 (1994), 819–825.CrossrefGoogle Scholar

  • [8]

    Dvurečenskij, A.—Pulmannová, S.: New Trends in Quantum Structures, Kluwer Acad. Pub., Dordrecht, 2000.Google Scholar

  • [9]

    Engesser, K.—Gabbay, D. M.—Lehmann, D.: Handbook of Quantum Logic and Quantum Structures, Elsevier, 2009.Google Scholar

  • [10]

    Ghirardato, P.—Marinacci, M.: Ambiguity made precise: A comparative foundation, J. Econom. Theory 102 (2002), 251–289.CrossrefGoogle Scholar

  • [11]

    Kagan, E.—Ben-Gal, I.: Navigation of quantum-controlled mobile robots. In: Recent Advances in Mobile Robotics (A. Topalov, ed.), In Tech, 2011, pp. 311–326.Google Scholar

  • [12]

    Kalmbach, G.: Orthomodular Lattices, Academic Press, London, 1983.Google Scholar

  • [13]

    Khare, M.—Roy, S.: Conditional entropy and the Rokhlin metric on an orthomodular lattice with Bayessian state, Internat. J. Theoret. Phys. 47 (2008), 1386–1396.CrossrefGoogle Scholar

  • [14]

    Khare, M.—Roy, S.: Entropy of quantum dynamical systems and sufficient families in orthomodular lattices with Bayessian state, Comm. Theor. Phys. 50 (2008), 551–556.CrossrefGoogle Scholar

  • [15]

    Khare, M.—Singh, A. K.: Atoms and Dobrakov submeasures in effect algebras, Fuzzy Sets and Systems 159 (2008), 1123–1128.CrossrefWeb of ScienceGoogle Scholar

  • [16]

    Khare, M.—Singh, A. K.: Atoms and Saks type decomposition in effect algebras, Novi Sad J. Math. 38 (2008), 59–70.Google Scholar

  • [17]

    Khare, M.—Singh, A. K.: Pseudo-atoms, atoms and a Jordan type decomposition in effect algebras, J. Math. Anal. Appl. 344 (2008), 238–252.CrossrefWeb of ScienceGoogle Scholar

  • [18]

    Khare, M.—Singh, A.K.: Weakly tight functions, their Jordan type decomposition and total variation in effect algebras, J. Math. Anal. Appl. 344 (2008), 535–545.Web of ScienceCrossrefGoogle Scholar

  • [19]

    Kolmogorov, A. N.: Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Berlin, 1933.Google Scholar

  • [20]

    Kôpka, F.—Chovanec: D-posets, Math. Slovaca 44 (1994), 21–34.Google Scholar

  • [21]

    Marczewski, E.: On compact measures, Fund. Math. 40 (1953), 113–124.CrossrefGoogle Scholar

  • [22]

    Pap, E.: Null-additive Set Functions, Kluwer Acad. Pub., Dordrecht, 1995.Google Scholar

  • [23]

    Shukla, A.: Theory of Generalized Measures and Applications: Generalized Measures on Quantum Structures and Entropy. D. Phil. Thesis, University of Allahabad, Allahabad, India, 2015.Google Scholar

  • [24]

    Solér, M. P.: Characterization of Hilbert space by orthomodular spaces, Comm. Algebra 23 (1995), 219–243.CrossrefGoogle Scholar

About the article

Received: 2016-08-26

Accepted: 2016-10-18

Published Online: 2018-05-18

Published in Print: 2018-06-26

Citation Information: Mathematica Slovaca, Volume 68, Issue 3, Pages 491–500, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0119.

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