In this paper, we first introduced the Riemann integral for an abstract-valued function from a finite closed real interval to a sequentially complete random Saks space and gave the fundamental theorem of calculus for an $L^0$-Lipschitz function. Then we investigated some important properties peculiar to bi-continuous semigroups on a sequentially complete random Saks space. Finally, based on the above work, we established the Hille-Yosida generation theorem for such bi-continuous semigroups, which extends and improves several known results.