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We suggest a new method of describing invariant measures on Markov compactaand path spaces of graphs, and thus of describing characters of some groups andtraces of AF-algebras. The method relies on properties of filtrationsassociated with the graph and, in particular, on the notion of a standardfiltration. The main tool is the so-called internal metric that we introduce onsimplices of measures; it is an iterated Kantorovich metric, and the key resultis that the relative compactness in this metric guarantees a constructiveenumeration of the ergodic invariant measures. Applications include a number ofclassical theorems on invariant measures

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