Two commuting weights in a von Neumann algebra

Question

Let $\varphi$ be a faithful weight on $M$. $\psi$ is a weight commuting with $\varphi$.

In Takesaki's book, Vol 2, Chapter VIII , Proposition 3.15. The author mentions that $\psi \varphi=\varphi \psi$ is equivalent to $\psi=\psi\circ \sigma_t^{\varphi}$.

How to derive the above conclusion?

Answer 1

The relation $\psi=\psi\circ\sigma_t^\varphi$ is how Takesaki defines "$\psi$ commuting with $\varphi$".