Decompositions Of T₀-Measures And T_∞-Measures

In 1991, Butnariu and Klement put forward an open problem as follows:Do there exist for finite T-measures Jordan decompositions by monotone T-measures when T is a Frank t-norm such that T T ?In 2001, Professor Zhang defined the inclusion variations, disjoint variations and chain variations of set functions, and then discussed the properties of the three kinds of variations. He found that variations play an important role in the decompositions of signed fuzzy measures and signed lower semicontinuous fuzzy measures, just as they do for signed measures, additive fuzzy measures, non-monotonic fuzzy measures and T -measures.As a result, we firstly define inclusion variations, disjoint variations and chain variations with respect to fuzzy set functions. Then, after a more systematic study, we show that variations with respect to fuzzy set functions have many similar properties to those of variations with respect to set functions. And one important conclusion is: A finite T0-measure defined on a generated Tx-tribe( [0, ]) has a Jordandecomposition if and only if it is of bounded chain variation, where T is a Frankt-norm.Thus, although we don't solve the open problem given by Butnariu and Klement in 1991, we really show that variations indeed play a role in the decompositions of To-measures.In the second part of our thesis, we discuss the decomposition theorems for particular T0-measures-Too-measures. As we all know, measures have three decomposition theorems in classical measure theory, that is, Hahn decomposition theorem, Jordan decomposition theorem and Lebesgue decomposition theorem. In 1978 and 1983, Schmidt and Butnariu prove the Hahn decomposition theorem for finite T -measures in a different way, respectively.In our paper, we firstly prove, in a similar way to the proof of Hahn decomposition theorem for signed measures, that Too-measures have Hahn decompositions, too. Then, we show that infinite T -measures just as finite Too-measures also have Jordan decompositions by using the corollary of Hahn decomposition theorem for T -measures. Lastly, by applying the representation theorem of T -measures, we get the Lebesgue decomposition theorem for Tco-measures.Moreover, for the convenience of those who have interest in this field, we list four problems in the last chapter of our paper.