  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
