Integral and ideals in Riesz spaces. (English) Zbl 1185.28016
The authors study a Bochner-type integral for Riesz space-valued function. They prove some Vitali and Lebesgue dominated convergence theorems. They present many particular cases.
Reviewer: Beloslav Riečan (Banská Bystrica)
MSC:
| 28B05 | Vector-valued set functions, measures and integrals |
Keywords:
Riesz spaces; ideals; Bochner integral; Vitali convergence theorems; Lebesgue dominated convergence theoremReferences:
| [1] | 177 (2) (2007) 655-656.; 177 (2) (2007) 655-656. · Zbl 1089.40001 |
| [2] | Aytar, S.; Pehlivan, S., Statistical cluster and extreme limit points of sequences of fuzzy numbers, Inform. Sci., 177, 16, 3290-3296 (2007) · Zbl 1121.40002 |
| [3] | Balcerzak, M.; Dems, K., Some types of convergence and related Baire systems, Real Anal. Exch., 30, 267-276 (2004/2005) · Zbl 1071.26003 |
| [4] | Balcerzak, M.; Dems, K.; Komisarski, A., Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl., 328, 715-729 (2007) · Zbl 1119.40002 |
| [5] | Boccuto, A., Riesz Spaces, Integration and Sandwich Theorems, vol. 3 (1993), Tatra Mountains Math. Publ., pp. 213-230 · Zbl 0815.28007 |
| [6] | Boccuto, A., Abstract Integration in Riesz Spaces, vol. 5 (1995), Tatra Mountains Math. Publ., pp. 107-124 · Zbl 0857.28007 |
| [7] | Boccuto, A., Integration in Riesz Spaces with respect to (D)-convergence, vol. 10 (1997), Tatra Mountains Math. Publ., pp. 33-54 · Zbl 0918.28010 |
| [8] | A. Boccuto, D. Candeloro, Itô formula in Riesz spaces, in: Proceedings of the 10th International Conference IPMU, Perugia, July 2004, pp. 1807-1814.; A. Boccuto, D. Candeloro, Itô formula in Riesz spaces, in: Proceedings of the 10th International Conference IPMU, Perugia, July 2004, pp. 1807-1814. |
| [9] | A. Boccuto, D. Candeloro, Kondurar theorem in Riesz spaces, in: Proceedings of the 10th International Conference IPMU, Perugia, July 2004, pp. 1815-1822.; A. Boccuto, D. Candeloro, Kondurar theorem in Riesz spaces, in: Proceedings of the 10th International Conference IPMU, Perugia, July 2004, pp. 1815-1822. |
| [10] | A. Boccuto, D. Candeloro, Defining Limits by means of Integrals, 2008, preprint.; A. Boccuto, D. Candeloro, Defining Limits by means of Integrals, 2008, preprint. · Zbl 1248.28018 |
| [11] | Boccuto, A.; Sambucini, A. R., On the De Giorgi-Letta integral with respect to means with values in Riesz spaces, Real Anal. Exch., 21, 793-810 (1995/96) · Zbl 0879.28019 |
| [12] | Boccuto, A.; Sambucini, A. R., Comparison between different types of abstract integrals in Riesz spaces, Rend. Circ. Mat. Palermo, Ser. II, 46, 255-278 (1997) · Zbl 0881.28008 |
| [13] | Demirci, K., \(I\)-limit superior and limit inferior, Math. Commun., 6, 165-172 (2001) · Zbl 0992.40002 |
| [14] | Dems, K., On \(I\)-Cauchy sequences, Real Anal. Exch., 30, 123-128 (2004/2005) · Zbl 1070.26003 |
| [15] | Dirik, F.; Aral, A.; Demirci, K., \(I\)-convergence of positive linear operators on \(L_p\) weighted spaces, J. Comput. Anal. Appl., 10, 75-81 (2008) · Zbl 1134.41004 |
| [16] | Duchoň, M.; Riečan, B., On the Kurzweil-Stieltjes Integral in Ordered Spaces, vol. 8 (1996), Tatra Mountains Math. Publ., pp. 133-141 · Zbl 0918.28013 |
| [17] | Fast, H., Sur la convergence statistique, Coll. Math., 2, 241-244 (1951) · Zbl 0044.33605 |
| [18] | Halmos, P. R., Measure theory (1950), D. Van Nostrand Company, Inc. · Zbl 0117.10502 |
| [19] | Kostyrko, P.; Mačaj, M.; Šalát, T., \(I\)-convergence and extremal \(I\)-limit points, Math. Slovaca, 55, 443-464 (2005) · Zbl 1113.40001 |
| [20] | Kostyrko, P.; Šalát, T.; Wilczyński, W., \(I\)-convergence, Real Anal. Exch., 26, 669-685 (2000/2001) · Zbl 1021.40001 |
| [21] | Lahiri, B. K.; Das, P., Further results on \(I\)-limit superior and limit inferior, Math. Commun., 8, 151-156 (2003) · Zbl 1043.40001 |
| [22] | Luxemburg, W. A.J.; Zaanen, A. C., Riesz Spaces I (1971), North-Holland Publishing Co. · Zbl 0231.46014 |
| [23] | Nihal Tuncer, A.; Berna Benli, F., \( \lambda \)-statistical limit points of the sequences of fuzzy numbers, Inform. Sci., 177, 160, 3297-3304 (2007) · Zbl 1122.40002 |
| [24] | Šalát, T.; Tripathy, B. C.; Ziman, M., On Some Properties of \(I\)-Convergence, vol. 28 (2004), Tatra Mountains Math. Publ., pp. 279-286 · Zbl 1110.40002 |
| [25] | Vulikh, B. Z., Introduction to the theory of partially ordered spaces (1967), Wolters-Noordhoff Sci. Publ. · Zbl 0186.44601 |
| [26] | Wagner, E., Sequences of measurable functions, Fund. Math., 112, 89-102 (1981) · Zbl 0386.28005 |
| [27] | Wagner-Bojakowska, E.; Wilczyński, W., Cauchy condition for the convergence in category, Proc. Am. Math. Soc., 128, 413-418 (1999) · Zbl 0932.28002 |
| [28] | Zaanen, A. C., Riesz Spaces II (1983), North-Holland Publishing Co. · Zbl 0519.46001 |
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