Why is $P_{\rho}$ is a probability measure on the Borel subsets of $H\ $?

I am going through a paper on Operator Probability Theory by Stan Gudder. The author introduced the notion of probability distribution of self-adjoint operators on a Hilbert space where the self-adjoint operators are thought of as complex valued random variables relative to a fixed state. Let $A \in \mathcal S (H)$ (self-adjoint operator) and $\rho$ be a state on $H.$ Let $P^A$ be the spectral measure corresponding to the self-adjoint operator $A.$ Then for a Borel subset $\Delta \subseteq \sigma (A)$ (spectrum of $A$) we define $$P_{\rho} (A \in \Delta) = \text{tr} \left (\rho P^A (\Delta) \right ).$$ So the expectation of $A$ is given as $:$$$E_{\rho} (A) = \int_{\sigma (A)} \lambda\ \text{tr} \left (\rho P^A (d \lambda) \right ) = \text{tr} (\rho A).$$

But I don't understand why $P_{\rho}$ is a valid probability measure. Also I don't follow why $E_{\rho} (A)$ evaluates to $\text {tr} (\rho A).$ Any suggestion in this regard would be warmly appreciated.

Thanks for your time.