Stochastic fuzzy analysis and its applications

以往对随机模糊集极限定理的研究主要是针对紧水平集的情形。这是由于它的方法是基于凸紧集在对偶空间中单位闭球上的有界连续函数空间中的嵌入定理。本文的目的是得到更广泛的随机模糊集的极限定理，推广了前人的研究。一种是有界但非紧集的情况，这种情况发生在无限维的情况下。另一个是无界集，这甚至发生在有限维的情况下。在前一种情况下，可能遇到的第一个问题是奥曼积分的封闭性。这个积分在随机集或随机模糊集的研究中是基本的，虽然已知一个关于封闭性的反例，但还不知道一个好的充分条件。在这个项目中，我们得到了Aumann积分是封闭的，如果基础空间 ...更多信息 e是自反Banach空间。接下来我们讨论收敛的概念，这在前一种情况和后一种情况下都很重要。在以往的工作中，模糊集被看作是水平集的简单集合，其拓扑结构有时与直觉不同。在这个项目中，我们引入了一个更符合直觉的新概念，称为图收敛，并改进了各种收敛定理。最后，我们还得到了图收敛的分离性定理，它保证了对于随机模糊集的收敛性证明，只需检验可数个水平集的收敛性.在此基础上，我们得到或改进了随机集和随机模糊集的各种极限定理，包括大数定律，鞅和s-鞅的极限定理，鞅的正则表示定理，随机模糊集的随机抽样定理，随机模糊集的逼近定理，高斯随机模糊集的表示定理。少

Former study on limit theorems for random fuzzy sets was mainly for the case of compact level sets. This is due to its method which is based on the embedding theorem of convex compact sets into the space of bounded continuous functions on the unit closed ball in the dual space. The object of this project is to obtain limit theorems for wider class of random fuzzy sets, extending the former study.There are two cases where the compactness condition fails. One is the case of bounded but non compact sets, which occurs in infinite dimension cases. The other is that of unbounded sets, which occurs even in finite dimensional cases. In the former case, the first problem one may encounter is the closedness of the Aumann integrals. This integral is fundamental in the study of random sets or random fuzzy sets, and although a counter example for the closeness was known, a good sufficient condition was not known. In this project, we obtained that the Aumann integral is closed if the underlying spac … More e is a reflexive Banach space. We next dealt with the concept of convergence, which is significant in both former and latter cases. In most of former works, the fuzzy sets were treated as simple collections of level sets and the topologies sometimes differs from intuition. In this project, we introduced a new concept called graph convergence which fits intuition more, and improved various convergence theorems. Finally, we also obtained separability theorem for graph convergence, which ensures that one have only to check the convergence of level sets for countably many levels for a proof of convergence of random fuzzy sets.Based on the former theoretical advance, we obtained or improved various limit theorems on random sets and/or random fuzzy sets, which covers law of large numbers, limit theorems for martingale and s-martingales, regular representation theorem for martingales, optional sampling theorems, approximation of random fuzzy sets and representation theorem of Gaussian random fuzzy sets. Less