A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting

References

[1] ALIPRANTIS, Ch. D.-BURKINSHAW, O.: Locally Solid Riesz Spaces with Applications to Economics (2nd ed.), Math. Surveys Monogr. 105, Amer. Math. Soc., Providence, RI, 2003.10.1090/surv/105Suche in Google Scholar

[2] AHMED, S. I.-PFEFFER, W. F.: A Riemann integral in a locally compact Hausdorff space, J. Aust. Math. Soc. Ser. A 41 (1986), 115-137.10.1017/S1446788700028123Suche in Google Scholar

[3] ABRAMOVICH,Yu.-SIROTKIN,G.: On order convergence of nets, Positivity 9 (2005), 287-292.10.1007/s11117-004-7543-xSuche in Google Scholar

[4] BOCCUTO, A.: Egorov property and weak σ-distributivity in (l)-groups, Acta Math. (Nitra) 6 (2003), 61-66.Suche in Google Scholar

[5] BOCCUTO, A.-DIMITRIOU, X.-PAPANASTASSIOU, N.: Basic matrix theorems for (I)-convergence in (l)-groups, Math. Slovaca 62 (2012), 269-298.Suche in Google Scholar

[6] 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/s12175-012-0070-5Suche in Google Scholar

[7] BOCCUTO, A.-MINOTTI, A. M.-SAMBUCINI, A. R.: Set-valued Kurzweil-Henstock integral in Riesz space setting, Panamer. Math. J. 23 (2013), 57-74.Suche in Google Scholar

[8] BOCCUTO, A.-RIEČAN, B.-VRÁBELOVÁ, M.: Kurzweil-Henstock Integral in Riesz Spaces, Bentham Sci. Publ., Sharjah, United Arab Emirates, 2009.Suche in Google Scholar

[9] BOCCUTO, A.-RIEČAN, B.: A note on the improper Kurzweil-Henstock integral, Acta Univ. M. Belii Ser. Math. 9 (2001), 7-12.Suche in Google Scholar

[10] BOCCUTO, A.-RIEČAN, B.: A note on improper Kurzweil-Henstock integral in Riesz spaces, Acta Math. (Nitra) 5 (2002), 15-24; Addendum to “A note on improper Kurzweil-Henstock integral in Riesz spaces”, Acta Math. (Nitra) 7 (2004), 53-59.Suche in Google Scholar

[11] BOCCUTO, A.-RIEČAN, B.: The concave integral with respect to Riesz space-valued capacities, Math. Slovaca 59 (2009), 647-660.10.2478/s12175-009-0153-0Suche in Google Scholar

[12] BOCCUTO, A.-RIEČAN, B.: On extension theorems for M-measures in (l)-groups, Math. Slovaca 60 (2010), 65-74.10.2478/s12175-009-0167-7Suche in Google Scholar

[13] BOCCUTO, A.-RIEČAN, B.-SAMBUCINI, A. R.: Convergence and Fubini Theorems for metric semigroup-valued functions defined on unbounded rectangles, Panamer. Math. J. 19 (2009), 1-12.Suche in Google Scholar

[14] BOCCUTO, A.-SAMBUCINI, A. R.: The Henstock-Kurzweil integral for functions defined on unbounded intervals and with values in Banach spaces, Acta Math. (Nitra) 7 (2004), 3-17. Suche in Google Scholar

[15] BOCCUTO, A.-SKVORTSOV, V. A.: Henstock-Kurzweil type integration of Riesz space-valued functions and applications to Walsh series, Real Anal. Exchange 29 (2003/2004), 419-438.10.14321/realanalexch.29.1.0419Suche in Google Scholar

[16] BOCCUTO, A.-SKVORTSOV, V. A.: On Kurzweil-Henstock type integrals with respect to abstract derivation bases for Riesz space-valued functions, J. Appl. Funct. Anal. 1 (2006), 251-270.Suche in Google Scholar

[17] FAURE, C. A.-MAWHIN, J.: The Hake’s property for some integrals over multidimensional intervals, Real Anal. Exchange 20 (1994/1995), 622-630.10.2307/44152544Suche in Google Scholar

[18] FREMLIN, D. H.: A direct proof of the Matthes-Wright integral extension theorem, J. Lond. Math. Soc. (2) 11 (1975), 276-284.10.1112/jlms/s2-11.3.276Suche in Google Scholar

[19] KURZWEIL, J.-JARNĺK, J.: Differentiability and integrabilty in n dimensions with respect to α-regular intervals, Results Math. 21 (1992), 138-151.10.1007/BF03323075Suche in Google Scholar

[20] LEE, P. Y.-VÝBORNÝ, R.: The Integral: An Easy Approach after Kurzweil and Henstock, Austral. Math. Soc. Lect. Ser. 14, Cambridge Univ. Press, Cambrige, 2000.Suche in Google Scholar

[21] LUKASHENKO, T. P.-SOLODOV, A.-SKVORTSOV, V. A.: Generalized integrals, Librokom, Moscow, 2010 (Russian).Suche in Google Scholar

[22] LUXEMBURG, W. A. J.-ZAANEN, A. C.: Riesz Spaces I, North-Holland Publ. Co., Amsterdam, 1971.Suche in Google Scholar

[23] MULDOWNEY, P.-SKVORTSOV, V. A.: Improper Riemann Integral and Henstock Integral in Rn, Mat. Zametki 78 (2005), 251-258 [Translation: Math. Notes 78 (2005), 228-233].Suche in Google Scholar

[24] OSTASZEWSKI, K. M.: Henstock integration in the plane, Mem. Amer. Math. Soc. 63 (1986), 1-353.Suche in Google Scholar

[25] PERESSINI, A. L.: Ordered Topological Vector Spaces, Harper & Row, New York, 1967.Suche in Google Scholar

[26] RIEČAN, B.-NEUBRUNN, T.: Integral, Measure and Ordering, Kluwer/Ister, Dordrecht/ Bratislava, 1997.10.1007/978-94-015-8919-2Suche in Google Scholar

[27] SAKS, S.: Theory of the Integral, Hafner Publishing Company, New York, 1937.Suche in Google Scholar

[28] SCHAEFER, H. H.: Banach Lattices and Positive Operators, Springer-Verlag, New York, 1974.10.1007/978-3-642-65970-6Suche in Google Scholar

[29] SKVORTSOV, V. A.-TULONE, F.: The P-integral in the theory of series with respect to characters of zero-dimensional groups, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 61 (2006), 25-29 [Translation: Moscow Univ. Math. Bull. 61 (2006), 27-31].Suche in Google Scholar

[30] SKVORTSOV, V. A.-TULONE, F.: Henstock-Kurzweil type integral on a compact zerodimensional metric space and representation of quasi-measure, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 67 (2012), 11-17 [Translation: Moscow Univ. Math. Bull. 67 (2012), 55-60].Suche in Google Scholar

[31] SKVORTSOV, V. A.-TULONE, F.: Generalized Hake property for integrals of Henstock type, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 68 (2013), 9-13 [Translation: Moscow Univ. Math. Bull. 68 (2013), 270-274].Suche in Google Scholar

[32] SWARTZ, C.: The Nikodým boundedness theorem for lattice-valued measures, Arch. Math. (Basel) 53 (1989), 390-393. 10.1007/BF01195219Suche in Google Scholar

[33] THOMSON, B. S.: Derivation bases on the real line, I; II, Real Anal. Exchange 8 (1982/83), 67-207; 278-442.10.2307/44153415Suche in Google Scholar

[34] TULONE, F.-ZHEREBYOV, Yu.: Absolutely continuous variational measures of Mawhin’s type, Math. Slovaca 61 (2011), 747-768.10.2478/s12175-011-0043-0Suche in Google Scholar

[35] VULIKH, B. Z.: Introduction to the theory of partially ordered spaces, Wolters-Noordhoff Sci. Publ., Groningen, 1967.Suche in Google Scholar

[36] WRIGHT, J. D. M.: The measure extension problem for vector lattices, Ann. Inst. Fourier (Grenoble) 21 (1971), 65-85. 10.5802/aif.393Suche in Google Scholar