Fuzzy set-valued stochastic Lebesgue integral

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closure is given, focusing on Aumann representation theorem, representation theorem and property of convexity. It is proved that the fuzzy closed set-valued stochastic Lebesgue integral is a fuzzy closed set-valued stochastic process which is widely used in the fuzzy world with randomness. Finally, the fuzzy closed set-valued stochastic Lebesgue integral in Lp-space is studied, especially on an almost everywhere problem.