We study the simplex of probability measures on a Bratteli diagram B which are invariant with respect to the tail equivalence relation. We prove a criterion of unique ergodicity of a Bratteli diagram. In case when a finite rank $k$ Bratteli diagram B has $1 \leq l \leq k$ ergodic invariant measures, we describe the structures of the diagram and the subdiagrams which support these measures.
This is a joint work with S. Bezuglyi and J. Kwiatkowski.