Trace of positive operator

Question

Let $A$ be a bounded $\ge 0$ operator on separable complex Hilbert space $\mathscr{H}$. By the spectral theorem, we know that there exists a unitary transform $U:\mathscr{H} \rightarrow L^2 (\mu)$ and finite measure $\mu$ and $f\in L^\infty (\mu),f\ge0$ such that \begin{equation} UAU^*\psi =f \psi \end{equation} Is it true that \begin{equation} \operatorname{tr}{A}=\int fd\mu \end{equation} How would one prove this?