In this paper we present a generalization of the Radon-Nikodym theorem proved by Pedersen and Takesaki. Given a normal, semifinite and faithful (n.s.f.) weight $��$ on a von Neumann algebra M and a strictly positive operator $��$, affiliated with M and satisfying a certain relative invariance property with respect to the modular automorphism group $��^��$ of $��$, with a strictly positive operator as the invariance factor, we construct the n.s.f. weight $��(��^{1/2} . ��^{1/2})$. All the n.s.f. weights on M whose modular automorphisms commute with $��^��$ are of this form, the invariance factor being affiliated with the centre of M. All the n.s.f. weights which are relatively invariant under $��^��$ are of this form, the invariance factor being a scalar.

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