Quasi-invariant measures for continuous group actions

The class of ergodic, invariant probability Borel measure for the shift action of a countable group is a G(delta) set in the compact, metrizable space of probability Borel measures. We study in this paper the descriptive complexity of the class of ergodic, quasi-invariant probability Borel measures and show that for any infinite countable group Gamma it is Pi(0)(3)-hard, for the group Z it is Pi(0)(3)-complete, while for the free group F-infinity with infinite, countably many generators it is Pi(0)(alpha)-complete, for some ordinal a with 3 <= alpha <= omega + 2. The exact value of this ordinal is unknown.