Construction and regularity of transition functions on polish spaces under measurability conditions

This paper concerns the construction and regularity of a transition (probability) function of a non-homogeneous continuous-time Markov process with given transition rates and a general state space. Motivating from a lot of restriction in applications of a transition function with continuous (in t ≥ 0) and conservative transition rates q(t, x, Λ), we consider the case that q(t, x, Λ) are only required to satisfy a mild measurability (in t ≥ 0) condition, which is a generalization of the continuity condition. Under the measurability condition we construct a transition function with the given transition rates, provide a necessary and sufficient condition for it to be regular, and further obtain some interesting additional results.