  In this paper the notion of Measure Equivalence (ME) of countable groups is
studied. ME was introduced by Gromov as a measure-theoretic analog of
quasi-isometries. All lattices in the same locally compact group are Measure
Equivalent; this is one of the motivations for this notion. The main result of
this paper is ME rigidity of higher rank lattices: any countable group which is
ME to a lattice in a simple Lie group G of higher rank, is commensurable to a
lattice in G.
