Two measures, m and m' on a topological space X are called homeomorphic if there is a homeomorphism f of X such that m(f(A)) = m'(A) for any Borel set A. The question when two Borel probability non-atomic measures are homeomorphic has a long history beginning with the work of Oxtoby and Ulam who gave a criterion when a probability Borel measure on the cube [0, 1]^n is homeomorphic to the Lebesgue measure. The situation is more interesting for measures on a Cantor set. There is no complete characterization of homeomorphic measures. But, for the class of the so called goodmeasures (introduced by E. Akin), the answer is simple: two good measures are homeomorphic if and only if the sets of their values on clopen sets are the same.
I will focus in the talk on the study of probability measures invariant with respect to a minimal (or aperiodic) homeomorphism. These measures are in one-to-one correspondence with traces on a corresponding dimension group. The technique of dimension groups allows us to apply new methods for studying good traces. A good trace is characterized by its kernel having dense image in the annihilating set of affine functions on the trace space. A number of examples with seemingly paradoxical properties is considered.
The talk will be based on a joint paper with D. Handelman.