Let $M$ be a von Neumann algebra and let $\varphi$ be a faithful normal semifinite weight on $M$. Let $N\subset M$ be a von Neumann subalgebra such that the restricted weight $\varphi|_N$ is semifinite and \begin{align*} \sigma_t^\varphi(N)=N \end{align*} for every $t\in\mathbb R$, where each $\sigma_t^\varphi:M\to M$ is a $*$-automorphism and $(\sigma_t^\varphi)_{t\in\mathbb R}$ is the modular automorphism group of $\varphi$. Then there exists a unique normal faithful [conditional expectation](/page/Conditional%20Expectation) $E:M\to N$ satisfying \begin{align*} \varphi(E(x))=\varphi(x) \end{align*} for every positive $x\in M$ for which the two sides are defined.