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Keywords:
Riesz spaces; Henstock-Kurzweil integral
Riesz spaces; Henstock-Kurzweil integral
Summary:
In this paper we introduce and investigate a Henstock-Kurzweil-type integral for Riesz-space-valued functions defined on (not necessarily bounded) subintervals of the extended real line. We prove some basic properties, among them the fact that our integral contains under suitable hypothesis the generalized Riemann integral and that every simple function which vanishes outside of a set of finite Lebesgue measure is integrable according to our definition, and in this case our integral coincides with the usual one.
References:
[1] A. Boccuto: Differential and integral calculus in Riesz spaces. Tatra Mt. Math. Publ. 14 (1998), 293–323. MR 1651221
[2] M. Duchoň and B. Riečan: On the Kurzweil-Stieltjes integral in ordered spaces. Tatra Mt. Math. Publ. 8 (1996), 133–141. MR 1475272
[3] D. H. Fremlin: Topological Riesz Spaces and Measure Theory. Cambridge Univ. Press, 1994. MR 0454575
[4] D. H. Fremlin: A direct proof of the Matthes-Wright integral extension theorem. J. London Math. Soc. 11 (1975), 276–284. MR 0380345 | Zbl 0313.06016
[5] L. P. Lee and R. Výborný: The integral: An easy approach after Kurzweil and Henstock. Cambridge Univ. Press, 2000. MR 1756319
[6] B. Riečan: On the Kurzweil integral for functions with values in ordered spaces I. Acta Math. Univ. Comenian. 56–57 (1990), 75–83. MR 1083014
[7] B. Riečan: On operator valued measures in lattice ordered groups. Atti Sem. Mat. Fis. Univ. Modena 41 (1993), 235–238. MR 1225686
[8] B. Riečan and T. Neubrunn: Integral, Measure and Ordering. Kluwer Academic Publishers/Ister Science, 1997. MR 1489521
[9] B. Riečan and M. Vrábelová: On the Kurzweil integral for functions with values in ordered spaces II. Math. Slovaca 43 (1993), 471–475. MR 1248980
[10] B. Riečan and M. Vrábelová: On integration with respect to operator valued measures in Riesz spaces. Tatra Mt. Math. Publ. 2 (1993), 149–165. MR 1251049
[11] B. Riečan and M. Vrábelová: On the Kurzweil integral for functions with values in ordered spaces III. Tatra Mt. Math. Publ. 8 (1996), 93–100. MR 1475267
[12] B. Riečan and M. Vrábelová: The Kurzweil construction of an integral in ordered spaces. Czechoslovak Math. J. 48(123) (1998), 565–574. DOI 10.1023/A:1022483929348 | MR 1637875
[13] J. D. M. Wright: The measure extension problem for vector lattices. Ann. Inst. Fourier Grenoble 21 (1971), 65–85. DOI 10.5802/aif.393 | MR 0330411 | Zbl 0215.48101
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