We study rigidity properties of lattices in Isom(Hn)=PO(n,1), n>2, and of surface groups in Isom(H2)=PGL(2,R) in the context of integrable Measure Equivalence. The results for lattices in Isom(Hn), n>2, are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable Measure Equivalence classification. For surface groups an integrable ME rigidity is obtained via a cocycle version of Milnor-Wood inequality. The integrability condition appears in a certain (co)homological tools pertaining to L1-homology and bounded cohomology of measure equivalent groups. Some of these homological tools are developed in a companion paper Efficient subdivision in hyperbolic groups and applications.