Measure classification is a central theme throughout ergodic theory. In this work, measure classification is given for actions of the following type: 1. Any unbounded group Г < G acting on GH where G = PSL2( R ) and H = Diff1(S1); 2. Any lattice Г < G acting on GH where G is as above and H = Homeo ac(S1), the group of absolutely continuous homeomorphisms of the circle; 3. Any uniform lattice Г < G where G is a connected, center free, real Lie group of rank one, acting on GH leaving some compact set Q invariant, where H = Homeo(G/P) where P < G is the unique (up to conjugation) proper parabolic subgroup of G; 4. Any real simple higher rankLie group G acting on H = Homeo(G/Q) where Q < G is any proper non-trivial parabolic subgroup.;For each of these cases, it turns out that the only invariant probability measure on X = GH is the Dirac measure on the identity coset. These results, though interesting in their own right, were motivated by studying embeddings of Teichmüller space Teich in the space of all (G, G)-couplings on H where G = PSL2( R ) and H = Homeo(S1). Additionally, these results are in part motivated by asking Ratner-type questions. Moreover, these results relate to a general classification of probability measures invariant under lattice actions on the spaces GH where H = Homeo(G/H) where G is any simple Lie group and Q < G is any proper parabolic subgroup, and, more generally, relate to classification of (G, G)-couplings on the spaces H described above.
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