Construction of a regular split process

In this paper, we develop the approach to the general theory of Markov processes proposed in [4]. Let $x_t$ be an (inhomogeneous) Markov process. The right regularization $x_{t+}$ and the left regularization $x_{t-}$ of the process $x_t$ are constructed. They have the following properties. Let $t$ be a real number and $A$ be an event belonging to the «future» $\mathscr F_{>t}$. Then, almost surely, the function $\mathbf P_{t+,x_{t+}}(A)$ is the right-continuous modification of $\mathbf P_{t-,x_{t-}}(A)$ and $\mathbf P_{t-,x_{t-}}(A)$ is the left-continuous modification of $\mathbf P_{t+,x_{t+}}(A)$, where $\mathbf P_{s+,x}$ (resp. $\mathbf P_{s-,x}$) are the transition probabilities of $x_{t+}$ (resp. $x_{t-}$).