This paper gives the Radon-Nikodym theorem in signed Loeb space under 1-saturated nonstandard model. First,the nonstandard characterization of absolute continuity is discussed,on which Radon-Nikodym theorem in signed Loeb space is obtained. Then,some facts about a finite signed Loeb measure and its variation are shown.
This paper, using the monads theory in the topological space, gives a new characterization of irreducible sets in the nonstandard enlarged models. Further, the discretization expression of Sober topological spaces is presented.
In this paper,the closeness of the τ-standard part of a set is discussed.Some related propositions of the τ-neighborhood system of a set are given.And then some related conclusions of the τ-monad of a set and the τ-standard part of a set are presented.And based on it,the necessary and sufficient conditions of the enlarged model and the saturated model are showed.Finally,some sufficient conditions that the τ-standard part of a set is closed are proved in the enlarged model and the saturated model.
The structure and some properties of ＊τx are discussed in the nonstandard κ-saturated model in this paper. First, a sufficient and necessary condition of a internal set in ＊τx is given. Then, in κ-saturation, some properties of ＊τx are proved. Finally, the approach theorem is easily obtained.
