The problem of constructing a strong Markov process with a given measurable Markov transition function $p(s,x,t,\Gamma )$ is considered. The space X of possible states is supposed to be given together with the function $p(s,x,t,\Gamma )$.If it is required that the sample functions $x(t,\omega )$ be defined for each $\omega $ at all $t \in [0,\infty )$, then as shown by the examples described, the three following types of measurable transition functions exist: 1) every Markov process with a given transition function is a strong Markov process; 2) there are both strong Markov processes, and measurable not strong Markov processes with a given transition function; 3) there are no strong Markov processes with a given transition function. To avoid the third case, it is important to know, whether there exists a process with a given transition function whose sample functions are continuous from the right. Under some general assumptions about the space X, a process with such sample functions exists if for each $s ...