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Conditions for Egoroff’s theorem in non-additive measure theory. (English) Zbl 1046.28013
Summary: This paper gives necessary and/or sufficient conditions for Egoroff ’s theorem in non-additive measure theory: a necessary and sufficient condition described without measurable functions, two sufficient conditions, and a necessary condition. One of the two sufficient conditions is strong order total continuity (continuity at measurable sets of measure zero with respect to net convergence), and the other is strong order continuity (sequential continuity at measurable sets of measure zero) together with property (S). The necessary condition is strong order continuity. In addition, the paper shows the following: continuity from above and below, which is a known sufficient condition, and the above-mentioned two sufficient conditions are independent of each other; the disjunction of these three sufficient conditions is not a necessary condition; if the underlying set is at most countable, then strong order continuity is necessary and sufficient; and generally, strong order continuity is not sufficient.
MSC:
28E10Fuzzy measure theory