A calculus for stochastic processes can be defined with the metric given by the Lebesgue norm on the probability space. This is a rich notion, distinct to pathwise operations, which is connected to the usual statistical moments and finds applications in the context of stochastic and random differential equations. In this paper, we want to advance in the theory of this random Lebesgue calculus. Specifically, we improve the existing literature on the topic by extending the random chain-rule theorem and the Leibniz integral rule to stochastic evaluation processes and stochastic integration limits.