Separability And Measurability For Interval-Valued Stochastic Processes

Fuzzy value and fuzzy value functions produce some concepts and theories which consists of fuzzy value metric space, varies of convergences, fuzzy value measurable functions, fuzzy value variables, fuzzy value function's Lebegue integral, and so on. According to mature theory frame of classic maths, fuzzy value and fuzzy value functions have large space to develop. Under this comparing, the concept of interval-value stochastic processes is defined and separability theorems and measurability theorems is proved.These questions are discussed in the article:1. Several concepts which are interval value , generalized interval value, generalized interval value's convergence, generalized interval value's metric space, generalized interval value measurable functions ,three convergences are defined and generalized interval value measurable functions' decomposed property and the relations among three convergences is proved. Finally, interval value Stochastic processes and its expectation are defined.2. Several concepts which are compact-metric space stochastic processes, random equivalence, separable set, the separability of compact-metric space stochastic processes are defined and separability theorems in compact-metri(?) space stochastic processes and interval value stochastic processes are proved.3. The measurability of interval value stochastic processes is defined and it is proved that if interval value stochastic processes is continuity ,it has measurability .4. Some properties of sample function are defined and it is proved that sample function is almost step function or continuous function in customizing conditions.In the future, fuzzy value stochastic processes will be important and interval value stochastic processes also need be investigated by more scholars, investigating some special interval value stochastic processes may be useful for practice needs.