The Egoroff Theorem for Riesz Space-valued Monotone Measures

In 1974, Sugeno introduced the notion of fuzzy measure and integral to evaluate nonadditive or non-linear quality in systems engineering. In the same year, Dobrakov independently introduced the notion of submeasure from mathematical point of view to show that most of the theory of countably additive measures remain valid for such measures. Fuzzy measures and submeasures are both special kinds of non-additive measures, and their studies have stimulated engineers’ and mathematicians’ interest in non-additive measure theory. The classical theorem of Egoroff is one of the most fundamental and important theorems in measure theory. Unfortunately, it is known that the Egoroff theorem does not remain valid in general within the framework of non-additive measure theory. Recently, Murofushi et al. [3] discovered a necessary and sufficient condition (the Egoroff condition) which assures that the Egoroff theorem is still valid for monotone measures, and indicated that the continuity from above and below is one of the sufficient conditions for the Egoroff condition. Those conditions can be naturally described for Riesz space-valued monotone measures. However, the ε-argument, which is useful in additive or non-additive measure theory, does not work in a general Riesz space, so that it is difficult to develop a theory of Riesz space-valued non-additive measures without any additional condition on the space. In this talk, instead of ε-argument, we introduce and impose a new smoothness condition (the asymptotic Egoroff property) on a Riesz space to show that the Egoroff theorem holds for any Riesz space-valued monotone measure which is continuous from above and below. Further, we remark that many important Riesz spaces, such as the space RS of all real functions on an arbitrary set S and the Lebesgue spaces Lp[0, 1] (0 < p ≤ ∞) have the asymptotic Egoroff property, although the space C[0, 1] does not have such a property. See [1, 2] for full details.