The Inclusion Variation of Non-additive Set Functions on T-Tribe

Abstract

Inclusion variation on T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> -tribe play an important role in T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> -measures and fuzzy games theory, where they were used for additive set functions on T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> -tribe. In this paper classical variations theory is extended to non-additive set functions on T-tribe, after defining the inclusion variation, we investigate its some properties in detailed, such as, the (null-) null-additivity, exhaustivity, order continuity, continuity from the left and so on. As to the inclusion variation on T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> -tribe, an alternative proof of the Jordan decomposition theorem is given. Some relative results about inclusion variation in classical measure theory are generalized.