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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 μ with values in a Riesz space V. If V is weakly sigma-distributive and μ is compact and uniformly autocontinuous, then μ 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 μ 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 μ.
28E10Fuzzy measure theory