Two measures, µ and ν on a topological space X are called homeomorphic if there is a homeomorphism f of X such that µ(f(A)) = ν(A) for any Borel set A. The question when two Borel probability non-atomic measures are homeomorphic has a long history. The well-known result of Oxtoby and Ulam gives a criterion when a Borel measure on the cube [0, 1]n is homeomorphic to the Lebesgue measure. The situation is more difficult for measures on a Cantor set. There is no complete answer to the above question even in the simplest case of Bernoulli trail measures. In my talk, I will discuss the recent results about classification of Borel probability measures which are ergodic and invariant with respect to aperiodic substitution dynamical systems. In other words, we consider the set M of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation. The properties of these measures related to the clopen values set S(µ) are studied. It is shown that for every measure µ ∈ M there exists a subgroup G ⊂ R such that S(µ) = G ∩ [0, 1], i.e. S(µ) is group-like. A criterion of goodness is proved for such measures. Based on this result, we classify the good measures from M up to a homeomorphism. It is proved that for every good measure µ ∈ M there exist countably many measures {µi}i∈N ⊂ M such that the measures µ and µi are homeomorphic but the tail equivalence relations on the corresponding Bratteli diagrams are not orbit equivalent.