Let $M$ be a smooth manifold, let $\mu$ be a positive smooth density on $M$, and let $m \in \mathbb{R}$. Suppose that $A \in \Psi_{\mathrm{cl}}^m(M)$ is a properly supported scalar classical pseudodifferential operator acting on $C_c^\infty(M)$. Let $A^*$ denote the formal adjoint of $A$ with respect to the pairing