In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process. Let E be a locally compact, separable, metric space. We denote by \mathcal E the Borel subsets of E. Let \Omega be the space of right continuous maps from [0,\infty) to E that have left limits in E, and for each, denote by X_t the coordinate map at t; for each , is the value of \omega at t. We denote the universal completion of \mathcal E by . For each , let and then, let For each Borel measurable function f on E, define, for each x \in E, Since and the mapping given by is right continuous, we see that for any uniformly continuous function f, we have the mapping given by is right continuous. Therefore, together with the monotone class theorem, for any universally measurable function f, the mapping given by, is jointly measurable, that is, measurable, and subsequently, the mapping is also -measurable for all finite measures \lambda on and \mu on. Here, is the completion of with respect to the product measure. Thus, for any bounded universally measurable function f on E, the mapping is Lebeague measurable, and hence, for each, one can define There is enough joint measurability to check that is a Markov resolvent on , which uniquely associated with the Markovian semigroup. Consequently, one may apply Fubini's theorem to see that The following are the defining properties of Borel right processes: