  In this paper, we consider an extension of the Poisson random measure for the
formulation of continuous-time reinforcement learning, such that both the
frequency and the width of the jumps depend on the path. Starting from a
general point process, we define a new Poisson random measure as limit of the
linear sum of these counting processes, and name it the Mesgaki random measure.
We also construct its Stochastic integral and It\^o's formula.
