Any Markov process in a Borel space has a transition function

The most natural way to define «Markov process» $x_t$ is to say that it is a stochastic process with the Markov property. However, in some of the most interesting applications it is possible to consider only the processes with transition function, which is a family $\{p_t^s(x,\Gamma)\}$ of conditional distributions of $x_t$ given $x_s$, satisfying Kolmogorov–Chapman equation $p_t^s p_u^t=p_u^s$, $s<t<u$. We prove that the Markov process has the transition function if its state space is universal (i. e. it is isomorphic to a universally measurable subset of a Polish space).