In this classic paper, Sakai proves the following Radon-Nikodym theorem:
Let $M$ be a von Neumann algebra, and let $\phi$ and $\psi$ be two normal positive linear functionals on $M$. If $\psi \leq \phi$, then there is a positive operator $t_0\in M$ such that $0 \leq t_0 \leq 1$, and $\psi(x) = \phi(t_0 x t_0)$ for all $x \in M$.
The paper provides no uniqueness result. One would naively expect that any two such operators $t_0$ and $t_1$ would satisfy $\phi((t_1-t_0)^2)=0$. I can find no such statement in the literature. Is this true?
Please note that $\phi$ is not assumed to be faithful.