Conditions for Egoroff's theorem in non-additive measure theory

• Published in: Fuzzy sets and systems Vol. 146; no. 1; pp. 135 - 146
• Main Authors: Murofushi, Toshiaki, Uchino, Kenta, Asahina, Shin
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.08.2004; Elsevier
• Subjects: Almost everywhere convergence; Almost uniform convergence; Applied sciences; Circuit properties; Computer science; control theory; systems; Control theory. Systems; Digital circuits; Electric, optical and optoelectronic circuits; Electronic circuits; Electronics; Exact sciences and technology; Information, signal and communications theory; Mathematical methods; Miscellaneous; Non-additive measures; System theory; Telecommunications and information theory; Theoretical computing; Almost everywhere convergence; Non-additive measures; Almost uniform convergence; Engineering; Measure theory; Necessary condition; Strong convergence; Fuzzy system; Sufficient condition; Necessary and sufficient condition; Sequential method; Information processing; Fuzzy set
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/j.fss.2003.09.006

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.