Equivariant Maps and Purely Atomic Spectrum

Let G be a connected semisimple Lie group with no compact factors and finite center and Γ ⊂ G an irreducible lattice. If (X1, μ1) and (X2, μ2) are G-spaces with finite invariant measures and the action of G has purely atomic spectrum (and is essentially free or essentially transitive) we prove that every measure preserving Γ-map φ: X1 → X2 is also a G-map.