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Given a unitary representation U of a locally compact abelian group G, we investigate the relationship between two cocycles a" a2: Vax = a2+ b for some unitary operator V commuting with U and some coboundary b. A necessary and sufficient condition is given in terms of canonical-finite measures defined on G -1. These results are applied to the representation of Z defined by the shift of a stationary Markov chain.doi:10.2307/2041869 fatcat:tsegrk3zmvdz5g7pyi45br4sa4