On Families of Random Transitions

It is important, for game-theoretical purposes to extend the notion of Markov transition probabilities to the case of consistent families of measures. To the families of random transitions is applicable the notion of consistency and hence, that of extensibility.
Theorem. {\it Regularity of the complex ${\mathfrak{K}}$ is the necessary and sufficient condition of extensibility of any consistent family of random transitions connected with the complex ${\mathfrak{K}}$.}
Some random transitions and their families can be obtained as limits of sequences of conditional probabilities obtained from consistent families of probability measures. The set of such families of random transitions is described and a criterion whether a family of random transitions belongs to this set is given.