Measures on effect algebras
-
Giuseppina Barbieri
,
Francisco J. García-Pacheco
Abstract
We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.
(Communicated by Anatolij Dvurečenskij)
Acknowledgement
The authors would like to express their deepest gratitude towards the referee for valuable comments and suggestions.
References
[1] Avallone, A.—Basile, A.: On a Marinacci uniqueness theorem for measures, J. Math. Anal. Appl. 286(2) (2003), 378–390.10.1016/S0022-247X(03)00274-9Search in Google Scholar
[2] Avallone, A.—Vitolo, P.: Decomposition and control theorems in effect algebras, Sci. Math. Jpn. 58(1) (2003), 1–14.Search in Google Scholar
[3] Aizpuru, A.—Moreno Pulido, S.—Rambla Barreno, F.: Phillips Lemma on effect algebras of sets, Math. Slovaca 63(3) (2013), 639–646.10.2478/s12175-013-0124-3Search in Google Scholar
[4] Aizpuru, A.—Tamayo, M.: Classical properties of measure theory on effect algebras, Fuzzy Sets and Systems 157(15) (2006), 2139–2143.10.1016/j.fss.2006.03.010Search in Google Scholar
[5] Barbieri, G.: Lyapunov’s theorem for measures on D-posets, Internat. J. Theoret. Phys. 43(7–8) (2004), 1613–1623.10.1023/B:IJTP.0000048807.37145.ccSearch in Google Scholar
[6] Barbieri, G.—Valente, A.—Weber, H.: Decomposition of ℓ-group-valued measures, Czechoslovak Math. J. 62(4) (2012), 1085–1100.10.1007/s10587-012-0065-ySearch in Google Scholar
[7] Bhaskara Rao, K. P. S.—Bhaskara Rao, M.: Theory of Charges. A Study of Finitely Additive Measures. Pure and Applied Mathematics 109, Academic Press, London, 1983.Search in Google Scholar
[8] Diestel, J.—Uhl, J. J.: Vector Measures, American Mathematical Society, Providence, R.I., 1977.10.1090/surv/015Search in Google Scholar
[9] Dunford, N.—Schwartz, J. T.: Linear Operators. Part I, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988.Search in Google Scholar
[10] Dvurečenskij, A.—Pulmannová, S.: New Trends in Quantum Structures. Mathematics and its Applications 516, Kluwer Academic Publishers, Dordrecht, Ister Science, Bratislava, 2000.10.1007/978-94-017-2422-7Search in Google Scholar
[11] Foulis, D. J.—Bennet, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. 24(10) (1994), 1331–1352.10.1007/BF02283036Search in Google Scholar
[12] Freniche, F. J.: The Vitali-Hahn-Saks theorem for Boolean algebras with the subsequential interpolation property, Proc. Amer. Math. Soc. 92(3) (1984), 362–366.10.1090/S0002-9939-1984-0759653-1Search in Google Scholar
[13] Hwang, H. T.—Li, L.—Kim, H.: Bounded vector measures on effect algebras, Bull. Austral. Math. Soc 72 (2005), 291–298.10.1017/S0004972700035085Search in Google Scholar
[14] Schachermayer, W.: On some classical measure-theoretic theorems for non–sigma–complete Boolean algebras, Dissertationes Math. (Rozprawy Mat.) 214 (1982), 33 pp.Search in Google Scholar
[15] Talagrand, M.: Propriété de Nikodým et propriété de Grothendieck, Studia Math. 78(2) (1984), 165–171.10.4064/sm-78-2-165-171Search in Google Scholar
[16] Weber, H.: Extension of modular functions and measures, Annali di Matematica (2017)10.1007/s10231-017-0703-ySearch in Google Scholar
[17] Wu, J.—Ma, Z.: The Brooks-Jewett theorem on effect algebras with the sequential completeness property, Czechoslovak J. Phys. 53 (2003), 379–383.10.1023/A:1024046900156Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Prof. RNDr. pavel brunovský, DrSc. passed away ∗dec. 5, 1934 – † dec. 14, 2018
- Ideals and congruences in pseudo-BCH algebras
- Regular double p-algebras
- Distributive nearlattices with a necessity modal operator
- States in generalized probabilistic models: An approach based in algebraic geometry
- Free power-associative n-ary groupoids
- On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions
- On the space of generalized theta-series for certain quadratic forms in any number of variables
- Uniqueness and periodicity for meromorphic functions with partial sharing values
- Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
- An inverse boundary problem for fourth-order Schrödinger equations with partial data
- Entropy as an integral operator
- Global behavior of two third order rational difference equations with quadratic terms
- Measures on effect algebras
- Some topological and combinatorial properties preserved by inverse limits
- The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces
- Compositions of porouscontinuous functions
- A note on prime divisors of polynomials P(Tk); k ≥ 1
- On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications
- Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations
