The problem of describing central measures on path space of graded graphs

We suggest a new method of describing invariant measures on Markov compacta and path spaces of graphs, and thus of describing characters of some groups and traces of AF-algebras. The method relies on properties of filtrations associated with the graph and, in particular, on the notion of a standard filtration. The main tool is the so-called internal metric that we introduce on simplices of measures; it is an iterated Kantorovich metric, and the key result is that the relative compactness in this metric guarantees a constructive enumeration of the ergodic invariant measures. Applications include a number of classical theorems on invariant measures