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 Riemann-integrable. 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.