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We define a general condition, called stability on extensions T of measure preserving transformations S . Stability is defined in terms of relative unique ergodicity, and as a joining property. Ergodic compact group extensions are stable, and moreover stable extensions satisfy lifting theorems similar to those satisfied by group extensions. In general, stable extensions have relative entropy zero. In the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensionsdoi:10.1090/s0002-9947-1992-1040044-2 fatcat:qx4bda2vvrg33g42ukywawpcdu