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The Alexandroff theorem for Riesz space-valued non-additive measures. (English) Zbl 1145.28012
The author discusses the counterparts of some classical results from the measure theory for the case of non-additive measure $\mu$ with values in a Riesz space $V$. If $V$ is weakly sigma-distributive and $\mu$ is compact and uniformly autocontinuous, then $\mu$ is continuous as well from above as from below, what is a version of the Alexandroff’s theorem. Similar results hold true if $V$ has the asymptotic Egoroff property and $\mu$ is autocontinuous. Moreover, Radon non-additive measures are examined, e.g., the equivalence of the Radon property and the regularity with tightness. The paper is an interesting contribution to the general measure theory not assuming additivity of the discussed measure $\mu$.

28E10Fuzzy measure theory
Full Text: DOI
[1] Alexandroff, A. D.: Additive set-functions in abstract spaces. Mat. sb. N.S. 9, No. 51, 563-628 (1941) · Zbl 67.0163.01
[2] Denneberg, D.: Non-additive measure and integral. (1997) · Zbl 0927.28011
[3] Dobrakov, I.; Farková, J.: On submeasures II. Math. slovaca 30, 65-81 (1980) · Zbl 0428.28001
[4] Fremlin, D. H.: A direct proof of the matthes -- wright integral extension theorem. J. London math. Soc. 2, No. 11, 276-284 (1975) · Zbl 0313.06016
[5] Hrachovina, E.: A generalization of the Kolmogorov consistency theorem for vector measures. Acta math. Univ. comenian. 54 -- 55, 141-145 (1988)
[6] Kawabe, J.: Uniformity for weak order convergence of Riesz space-valued measures. Bull. austral. Math. soc. 71, 265-274 (2005) · Zbl 1076.28003
[7] Kawabe, J.: The egoroff theorem for non-additive measures in Riesz spaces. Fuzzy sets and systems 157, 2762-2770 (2006) · Zbl 1106.28005
[8] Kawabe, J.: The egoroff property and the egoroff theorem in Riesz space-valued non-additive measure theory. Fuzzy sets and systems 158, 50-57 (2007) · Zbl 1117.28013
[9] Kawabe, J.: Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures. Fuzzy sets and systems 158, 895-903 (2007) · Zbl 1121.28021
[10] Li, J.: Order continuous of monotone set function and convergence of measurable functions sequence. Appl. math. Comput. 135, 211-218 (2003) · Zbl 1025.28012
[11] Li, J.; Yasuda, M.: Lusin’s theorem on fuzzy measure spaces. Fuzzy sets and systems 146, 121-133 (2004) · Zbl 1046.28012
[12] Li, J.; Yasuda, M.; Song, J.: Regularity properties of null-additive fuzzy measure on metric spaces. Lecture notes in artificial intelligence 3558, 59-66 (2005) · Zbl 1121.28305
[13] Luxemburg, W. A. J.; Zaanen, A. C.: Riesz spaces I. (1971) · Zbl 0231.46014
[14] Marczewski, E.: On compact measures. Fund. math. 40, 113-124 (1953) · Zbl 0052.04902
[15] Pap, E.: Null-additive set functions. (1995) · Zbl 0856.28001
[16] Parthasarathy, K. R.: Probability measures on metric spaces. (1967) · Zbl 0153.19101
[17] Pfanzagl, J.; Pierlo, W.: Compact systems of sets. Lecture notes in mathematics 16 (1966) · Zbl 0161.36604
[18] Riečan, J.: On the Kolmogorov consistency theorem for Riesz space valued measures. Acta math. Univ. comenian. 48 -- 49, 173-180 (1986)
[19] Riečan, B.; Neubrunn, T.: Integral, measure, and ordering. (1997) · Zbl 0916.28001
[20] Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures. (1973) · Zbl 0298.28001
[21] Volauf, P.: Alexandrov and Kolmogorov consistency theorem for measures with values in partially ordered groups. Tatra mt. Math. publ. 3, 237-244 (1993) · Zbl 0820.28006
[22] Wang, Z.; Klir, G. J.: Fuzzy measure theory. (1992) · Zbl 0812.28010
[23] Wright, J. D. M.: The measure extension problem for vector lattices. Ann. inst. Fourier Grenoble 21, 65-85 (1971) · Zbl 0215.48101
[24] Wu, C.; Ha, M.: On the regularity of the fuzzy measure on metric fuzzy measure spaces. Fuzzy sets and systems 66, 373-379 (1994) · Zbl 0844.28009
[25] Wu, J.; Wu, C.: Fuzzy regular measures on topological spaces. Fuzzy sets and systems 119, 529-533 (2001) · Zbl 0983.28009