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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 64, Issue 3


On the Kluvánek construction of the Lebesgue integral with respect to a vector measure

Beloslav Riečan
  • Department of Mathematics Faculty of Natural Sciences, Matej Bel University, Tajovského 40, SK-974 01, Banská Bystrica, Slovakia
  • Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-841 01, Bratislava, Slovakia
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Published Online: 2014-07-05 | DOI: https://doi.org/10.2478/s12175-014-0236-4


The Kluvánek construction of the Lebesgue integral is extended in two directions. First, instead of a compact interval [a, b] in the real line an abstract non-empty set X is considered, instead of the ring generated by subintervals of [a, b] an arbitrary ring A of subsets of X. Secondly, instead of the length of intervals (λ([c, d]) = d−c) any vector measure λ: A→V is considered, where V is a Riesz space.

MSC: Primary 28B05; 28B10; 28B15

Keywords: Lebesgue integral; Riesz space

  • [1] BOCCUTO, A.: Abstract integration in Riesz spaces, Tatra Mt. Math. Publ. 5 (1995), 107–124. Google Scholar

  • [2] BOCCUTO, A.— RIEČAN, B.— VRÁBELOVÁ, M.: Kurzweil-Henstock integral in Riesz spaces, Bentham Sci. Publ., Oak Park, IL, 2009. Google Scholar

  • [3] CHOVANEC, F.: States and observables on MV-algebras, Tatra. Mt. Math. Publ. 3 (1993), 55–65. Google Scholar

  • [4] CRISTESCU, R.: On integration in ordered vector spaces and on some linear operators, Rend. Circ. Mat. Palermo (2) Suppl. 33 (1993), 289–299. Google Scholar

  • [5] DVUREČENSKIJ, A.— PULMANNOVÁ, S.: New Trends in Quantum Structures, Kluwer Acad. Publ./Ister Sci., Dordrecht/Bratislava, 2000. http://dx.doi.org/10.1007/978-94-017-2422-7CrossrefGoogle Scholar

  • [6] FREMLIN, D. H.: Topological Riesz Spaces and Measure Theory, Cambridge Univ. Press, London, 1974. http://dx.doi.org/10.1017/CBO9780511897207CrossrefGoogle Scholar

  • [7] FREMLIN, D. H.: Measure Algebras, Handbook of Boolean algebras. North Holland, Amsterdam, 1989. Google Scholar

  • [8] JAKUBÍK, J.: Weak σ-distributivity of lattice ordered groups, Math. Bohem. 126 (2001), 151–159. Google Scholar

  • [9] JAKUBÍK, J.: On complete MV-algebras, Czechoslovak Math. J. 45 (1995), 473–480. Google Scholar

  • [10] KLUVÁNEK, I.: Archimedes was right, Elem. Math. 42 (1987), 51–114. Google Scholar

  • [11] KLUVÁNEK, I.: Integral calculus of functions of one real variable, PFUK Ružomberok, 2008 (Slovak). Google Scholar

  • [12] KÔPKA, F.: Boolean D-posets as the factor spaces, Internat. J. Theoret. Phys. 37 (1998), 93–101. http://dx.doi.org/10.1023/A:1026665306697CrossrefGoogle Scholar

  • [13] MALIČKÝ, P.: Random variables with values in a vector lattice, Acta Math. Univ. Comenian. 52–53 (1987), 249–263. Google Scholar

  • [14] MESIAR, R.— KOMORNÍKOVÁ, M.: Probability measures on interval-valued fuzzy events, Acta Univ. M. Belii Ser. Math. 19 (2011), 5–10. Google Scholar

  • [15] MUNDICI, D.: Advanced Lukasiewicz Calculus and MV-algebras, Springer, Dordrecht, 2011. http://dx.doi.org/10.1007/978-94-007-0840-2Web of ScienceCrossrefGoogle Scholar

  • [16] POTOCKÝ, R.: On the expected value of vector lattice — valued random variables, Math. Slovaca 36 (1986), 889–894. Google Scholar

  • [17] MONTAGNA, F.: An algebraic approach to propositional fuzzy logic, J. Log. Lang. Inf. 9 (2000), 91–1214. http://dx.doi.org/10.1023/A:1008322226835CrossrefGoogle Scholar

  • [18] RIEČAN, B.: On the product MV-algebras, Tatra Mt. Math. Publ. 16 (1999), 143–149. Google Scholar

  • [19] RIEČAN, B.: Probability theory on IF events. In: Algebraic and Proof-Theoretic Aspects of Non-classical Logics, Springer, Heidelberg, 2007, pp. 290–308. Google Scholar

  • [20] RIEČAN, B.: Analysis of fuzzy logic models. In: Intelligent Systems (V. M. Koleshko, ed.), INTECH, 2012, pp. 217–244. Google Scholar

  • [21] RIEČAN, B.: On the Kluvánek construction of the Lebesgue integral, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. (To appear). Google Scholar

  • [22] RIEČAN, B.— MUNDICI, D.: Probability on MV-algebras. In: Handbook of Measure Theory (E. Pap, ed.), Elsevier Science, Amsterdam, 2002, pp. 869–909. Google Scholar

  • [23] RIEČAN, B.— NEUBRUNN, T.: Integral, Measure, and Ordering, Kluwer, Dordrecht, 1997. Google Scholar

  • [24] RIEČAN, B.— TKÁČIK, Š.: A note on the Kluvánek integral, Tatra Mt. Math. Publ. 49 (2011), 1–7. Google Scholar

  • [25] VRÁBELOVÁ, M.: On the extension of subadditive measures in lattice ordered groups, Czechoslovak Math. J. 57 (2007), 95–103. http://dx.doi.org/10.1007/s10587-007-0046-8Web of ScienceCrossrefGoogle Scholar

About the article

Published Online: 2014-07-05

Published in Print: 2014-06-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 3, Pages 727–740, ISSN (Online) 1337-2211, DOI: https://doi.org/10.2478/s12175-014-0236-4.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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