Limit Theorems for k-Subadditive Lattice Group-Valued Capacities in The Filter Convergence Setting

Boccuto, Antonio; Dimitriou, Xenofon

doi:10.1515/tmmp-2016-0001

Abstract

We investigate some properties of lattice group-valued positive, monotone and k-subadditive set functions, and in particular, we give some comparisons between regularity and continuity from above. Moreover, we prove different kinds of limit theorems with respect to filter convergence. Furthermore, some open problems are posed.

References

[1] ATANASSOV, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Physica-Verlag, Heidelberg, 1999.10.1007/978-3-7908-1870-3
Search in Google Scholar
[2] AVILES LOPEZ, A.-CASCALES SALINAS, B.-KADETS, V.-LEONOV, A.: The Schur ℓ1 theorem for filters, J. Math. Phys., Anal., Geom. 3 (2007), 383-398.
Search in Google Scholar
[3] BERNAU, S. J.: Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. 15 (1965), 599-631.10.1112/plms/s3-15.1.599
Search in Google Scholar
[4] BOCCUTO, A.: Egorov property and weak σ-distributivity in Riesz spaces, Acta Math. (Nitra) 6 (2003), 61-66.
Search in Google Scholar
[5] BOCCUTO, A.-DIMITRIOU, X.: Ideal limit theorems and their equivalence in (ℓ)-group setting, J. Math. Res. 5 (2013), 43-60.10.5539/jmr.v5n2p42
Search in Google Scholar
[6] BOCCUTO, A.-DIMITRIOU, X.: Convergence Theorems for Lattice Group-Valued Measures. Bentham Sci. Publ., U. A. E., 2015.10.2174/97816810800931150101
Search in Google Scholar
[7] BOCCUTO, A.-DIMITRIOU, X.-PAPANASTASSIOU, N.: Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in (ℓ)-groups, Tatra Mt. Math. Publ. 49 (2011), 17-26.
Search in Google Scholar
[8] BOCCUTO, A.-DIMITRIOU, X.-PAPANASTASSIOU,N.: Some versions of limit and Dieudonn´e-type theorems with respect to filter convergence for (ℓ)-group-valued measures, Cent. Eur. J. Math. 9 (2011), 1298-1311.10.2478/s11533-011-0083-2
Search in Google Scholar
[9] BOCCUTO, A.-DIMITRIOU, X.-PAPANASTASSIOU, N.: Schur lemma and limit theorems in lattice groups with respect to filters, Math. Slovaca 62 (2012), 1145-1166.10.2478/s12175-012-0070-5
Search in Google Scholar
[10] BOCCUTO, A.-MINOTTI, A. M.-SAMBUCINI, A. R.: Set-valued Kurzweil-Henstock integral in Riesz space setting, PanAmer. Math. J. 23 (2013), 57-74.
Search in Google Scholar
[11] BOCCUTO, A.-RIEČAN, B.-SAMBUCINI, A. R.: On the product of M-measures in (ℓ)-groups, Aust. J. Math. Anal. Appl. 7 (2010), Article No. 9, 8 p.
Search in Google Scholar
[12] BOCCUTO, A.-RIEˇCAN, B.-VR´ABELOV´A, M.: Kurzweil-Henstock Integral in Riesz Spaces. Bentham Sci. Publ., U. A. E., 2009.
Search in Google Scholar
[13] BOCCUTO, A.-SAMBUCINI, A. R.: The monotone integral with respect to Riesz spacevalued capacities, Rend. Mat. Appl. 16 (1996), 491-524.
Search in Google Scholar
[14] CANDELORO, D.-SAMBUCINI, A. R.: Filter convergence and decompositions for vector lattice-valued measures, Mediterr. J. Math. 12 (2015), 612-637.10.1007/s00009-014-0431-0
Search in Google Scholar
[15] CAVALIERE, P.-DE LUCIA, P.: On Brooks-Jewett, Vitali-Hahn-Saks and Nikod´ym convergence theorems for quasi-triangular functions, Rend. Lincei Mat. Appl. 20 (2009), 387-396.
Search in Google Scholar
[16] DOBRAKOV, I.: On submeasures I, Dissertationes Math. 112 (1974), 5-35.
Search in Google Scholar
[17] DOBRAKOV, I.-FARKOV´A, J.: On submeasures II, Math. Slovaca 30 (1980), 65-81.
Search in Google Scholar
[18] DREWNOWSKI, L.: Topological rings of sets, Continuous set functions, Integration. I, II, III, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 20 (1972), 269-276, 277-286, 439-445.
Search in Google Scholar
[19] FREMLIN, D. H.: A direct proof of the Matthes-Wright integral extension theorem, J. London Math. Soc. 11 (1975), 276-284.10.1112/jlms/s2-11.3.276
Search in Google Scholar
[20] GUARIGLIA, E.: Uniform boundedness theorems for k-triangular set functions, Acta Sci. Math. 54 (1990), 391-407.
Search in Google Scholar
[21] PAP, E.: A generalization of a Dieudonn´e theorem for k-triangular set functions, Acta Sci. Math. 50 (1986), 159-167.
Search in Google Scholar
[22] PAP, E.: The Vitali-Hahn-Saks Theorems for k-triangular set functions, Atti Sem. Mat. Fis. Univ. Modena 35 (1987), 21-32.
Search in Google Scholar
[23] PAP, E.: Null-Additive Set Functions, in: Math. Appl., Vol. 337, Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava, 1995.
Search in Google Scholar
[24] RIEˇCAN, B.-NEUBRUNN, T.: Integral, Measure and Ordering, in: Math. Appl., Vol. 411, Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava, 1997.10.1007/978-94-015-8919-2
Search in Google Scholar
[25] SAEKI, S.: The Vitali-Hahn-Saks theorem and measuroids, Proc. Amer. Math. Soc. 114 (1992), 775-782.10.1090/S0002-9939-1992-1088446-8
Search in Google Scholar
[26] VENTRIGLIA, F.: Dieudonn´e’s theorem for non-additive set functions, J. Math. Anal. Appl. 367 (2010), 296-303.10.1016/j.jmaa.2010.01.017
Search in Google Scholar

Limit Theorems for k-Subadditive Lattice Group-Valued Capacities in The Filter Convergence Setting

Abstract

Paradigm

My account