Some properties of the variations of non-additive set functions on T-tribes

The inclusion variation and chain variation on T∞-tribes play an important role in T∞-measures and fuzzy games theory, where they were used for additive set functions on T∞-tribe. In this paper classical variation theory is extended to non-additive set functions on T-tribes, after defining the inclusion variation, disjoint and chain variation on T-tribes, we investigate some properties of these three variations in detail, such as, the (null-) null-additivity, exhaustivity, order continuity, continuity from the left and so on. An alternative proof of the Jordan decomposition theorem is given. Thus, some results pertaining to variations in classical measure theory are generalized. In some case, as for T∞-measures on T∞-tribe, its different formal variations can be extended to Ts-measures on Ts-tribes.