A Radon-Nikodym theorem for von Neumann algebras
  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 $\phi$ on a von Neumann algebra M and a strictly positive operator $\delta$, affiliated with M and satisfying a certain relative invariance property with respect to the modular automorphism group $\sigma^\phi$ of $\phi$, with a strictly positive operator as the invariance factor, we construct the n.s.f. weight $\phi(\delta^{1/2} . \delta^{1/2})$. All the n.s.f. weights on M whose modular automorphisms commute with $\sigma^\phi$ 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 $\sigma^\phi$ are of this form, the invariance factor being a scalar. 
Stefaan Vaes