关于从闭区间到完备随机赋范模的抽象值函数的Riemann 可积性的进一步研究

郭铁信和张霞最近引入和研究了从一个闭区间到一个完备随机赋范模的抽象值函数的Riemann积分, 证明了值域几乎处处有界的连续函数是Riemann 可积的. 本文首先给出该结果的一个更简短的证明, 使得我们对于值域的几乎处处有界性有一个更深的认识, 受此启发, 我们进一步构造两个例子, 其一说明值域并非几乎处处有界的连续函数也可以是Riemann 可积的, 另一例子说明连续函数可以非Riemann 可积. 最后, 我们证明从一闭区间到一个满支撑的完备随机赋范模的所有连续函数都Riemann 可积的充要条件是基底概率空间本质上由至多可数原子生成.

A further study on the Riemann-integrability for abstract-valued functions from a closed real interval to a complete random normed module

Guo and Zhang lately introduced and investigated the Riemann integral for abstract-valued functions from a closed real interval to a complete random normed module, they proved that a continuous function whose range is almost surely bounded is Riemanmintegrable. In this paper, we first give a more concise proof of their result, leading us to a further understanding of almost surely boundedness of range. Then, we are inspired to construct two examples, one of which suggests that a continuous function whose range is not almost surely bounded may also be Riemann-integrable, while the other shows a continuous function may be not Riemann-integrable. Finally, we prove that all continuous functions from a fixed closed interval to a given complete random normed module with full support are Riemann-integrable if and only if the base probability space is essentially generated by at most countably many atoms.