复完备随机内积模上的随机酉算子群的Stone表示定理

Stone’s representation theorem of a group of random unitary operators on complete complex random inner product modules

设[a,b]是有限实区间,(S,∥·∥)是完备的随机赋范模并赋予(ε,λ)-拓扑.在本文中,我们首先引进了从[a,b]到S的抽象值函数的Riemann积分并给出值域几乎处处有界的连续函数Riemann可积的一个充分条件.然后我们研究了随机谱测度和随机测度之间的关系.最后,在上述两个准备工作的基础之上,我们建立了复完备随机内积模上随机酉算子群的Stone表示定理.

Let [a,b] be a finite real interval and （S, ‖ ‖） a complete random normed module endowed with the （ε, λ）-topology. We first introduce the Riemann integral for abstract-valued functions from [a, b] to S and give a sufficient condition for such a continuous function with the almost surely bounded range to be Riemann- integrable. Then we investigate the relation between random spectral measures and ordinary random measures. Finally, based on the above two preliminaries, we establish the Stone＇s representation theorem of a group of random unitary operators on complete complex random inner product modules.