Article

 Title: Lattice-valued Borel measures. III.  (English) Author: Khurana, Surjit Singh Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 44 Issue: 4 Year: 2008 Pages: 307-316 Summary lang: English . Category: math . Summary: Let $X$ be a completely regular $T_{1}$ space, $E$ a boundedly complete vector lattice, $C(X)$ $(C_{b}(X))$ the space of all (all, bounded), real-valued continuous functions on $X$. In order convergence, we consider $E$-valued, order-bounded, $\sigma$-additive, $\tau$-additive, and tight measures on X and prove some order-theoretic and topological properties of these measures. Also for an order-bounded, $E$-valued (for some special $E$) linear map on $C(X)$, a measure representation result is proved. In case $E_{n}^{*}$ separates the points of $E$, an Alexanderov’s type theorem is proved for a sequence of $\sigma$-additive measures. Keyword: order convergence Keyword: tight and $\tau$-smooth lattice-valued vector measures Keyword: measure representation of positive linear operators Keyword: Alexandrov’s theorem MSC: 28A33 MSC: 28B15 MSC: 28C05 MSC: 28C15 MSC: 46B42 MSC: 46G10 idZBL: Zbl pre05819010 idMR: MR2493427 . Date available: 2009-01-29T09:15:33Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/119770 . Reference: [1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators.Academic Press, 1985. Zbl 0608.47039, MR 0809372 Reference: [2] Diestel, J., Uhl, J. J.: Vector measures.Math. Surveys 15 (1977), 322. Zbl 0369.46039, MR 0453964 Reference: [3] Kaplan, S.: The second dual of the space of continuous function.Trans. Amer. Math. Soc. 86 (1957), 70–90. MR 0090774 Reference: [4] Kaplan, S.: The second dual of the space of continuous functions IV.Trans. Amer. Math. Soc. 113 (1964), 517–546. Zbl 0126.12002, MR 0170205 Reference: [5] Kawabe, J.: The Portmanteau theorem for Dedekind complete Riesz space-valued measures.Nonlinear Analysis and Convex Analysis, Yokohama Publ., 2004, pp. 149–158. Zbl 1076.28004, MR 2144038 Reference: [6] Kawabe, J.: Uniformity for weak order convergence of Riesz space-valued measures.Bull. Austral. Math. Soc. 71 (2) (2005), 265–274. MR 2133410 Reference: [7] Khurana, Surjit Singh: Lattice-valued Borel Measures.Rocky Mountain J. Math. 6 (1976), 377–382. MR 0399409 Reference: [8] Khurana, Surjit Singh: Lattice-valued Borel Measures II.Trans. Amer. Math. Soc. 235 (1978), 205–211. Zbl 0325.28012, MR 0460590 Reference: [9] Khurana, Surjit Singh: Vector measures on topological spaces.Georgian Math. J. 14 (2007), 687–698. Zbl 1154.46025, MR 2389030 Reference: [10] Kluvanek, I., Knowles, G.: Vector measures and Control Systems.North-Holland Math. Stud. 20 (58) (1975), ix+180 pp. MR 0499068 Reference: [11] Lewis, D. R.: Integration with respect to vector measures.Pacific J. Math. 33 (1970), 157–165. Zbl 0195.14303, MR 0259064 Reference: [12] Lipecki, Z.: Riesz representation representation theorems for positive operators.Math. Nachr. 131 (1987), 351–356. MR 0908823 Reference: [13] Meyer-Nieberg, P.: Banach Lattices and positive operators.Springer-Verlag, 1991. MR 1128093 Reference: [14] Schaefer, H. H.: Banach Lattices and Positive Operators.Springer-Verlag, 1974. Zbl 0296.47023, MR 0423039 Reference: [15] Schaefer, H. H.: Topological Vector Spaces.Springer-Verlag, 1986. MR 0342978 Reference: [16] Schaefer, H. H., Zhang, Xaio-Dong: A note on order-bounded vector measures.Arch. Math. (Basel) 63 (2) (1994), 152–157. MR 1289297 Reference: [17] Schmidt, K. D.: On the Jordan decomposition for vector measures. Probability in Banach spaces, IV.(Oberwolfach 1982) Lecture Notes in Math. 990 (1983), 198–203, Springer, Berlin-New York. MR 0707518 Reference: [18] Schmidt, K. D.: Decompositions of vector measures in Riesz spaces and Banach lattices.Proc. Edinburgh Math. Soc. (2) 29 (1) (1986), 23–39. Zbl 0569.28011, MR 0829177 Reference: [19] Varadarajan, V. S.: Measures on topological spaces.Amer. Math. Soc. Transl. Ser. 2 48 (1965), 161–220. Reference: [20] Wheeler, R. F.: Survey of Baire measures and strict topologies.Exposition. Math. 2 (1983), 97–190. Zbl 0522.28009, MR 0710569 Reference: [21] Wright, J. D. M.: Stone-algebra-valued measures and integrals.Proc. London Math. Soc. (3) 19 (1969), 107–122. Zbl 0186.46504, MR 0240276 Reference: [22] Wright, J. D. M.: The measure extension problem for vector lattices.Ann. Inst. Fourier (Grenoble) 21 (1971), 65–85. Zbl 0215.48101, MR 0330411 Reference: [23] Wright, J. D. M.: Vector lattice measures on locally compact spaces.Math. Z. 120 (1971), 193–203. Zbl 0198.47803, MR 0293373 Reference: [24] Wright, J. D. M.: Measures with values in partially ordered vector spaces.Proc. London Math. Soc. 25 (1972), 675–688. MR 0344413 Reference: [25] Wright, J. D. M.: An algebraic characterization of vector lattices with Borel regularity property.J. London Math. Soc. 7 (1973), 277–285. MR 0333116 .

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