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Let (X¡, 2(, ji,) and (X2, 22, ji^) be two o-finite measure spaces. We show that any isomorphism 7* of the Banach space L\XX, 21; /i,) onto the Banach space Ll(X2, 22> lh) which satisfies ||r|| ||r-l|| < 2 induces a transformation of the underlying measure spaces. In  and  it has been shown by D. Amir and M. Cambern that if Yx and Y2 are compact Hausdorff spaces, and if there exists an isomorphism T of C( Yx) onto C(Y2) with ||F|| ||F_1|| < 2, then Yx and Y2 are homeomorphic. In thisdoi:10.1090/s0002-9939-1980-0550500-6 fatcat:5rzk4vcc7zdrdiuyrleferzfly