I'm reading an article about environmental monitoring and information transfer. Suppose $S$ represents a quantum system and $E$ is the environment. Assume at time $t=0$ there are no correlations between $S$ and $E$: $\rho_{SE}(0)=\rho_{S}(0)\otimes\rho_{E}(0)$, and this composite density operator evolved under the action of $U(t) = e^{-iHt/h}$, where $H$ is the total Hamiltonian. Let $P_\alpha$ be a projective operator on $E$. Then, the probability of obtaining outcome $α$ in this measurement when $S$ is described by the density operator $\rho_s(t)$ is given as
$$ \text{Prob}(\alpha|\rho_s(t))=\text{Tr}_E (P_αρ_E(t)) $$
and the density matrix of $S$ conditioned on the particular outcome $\alpha$ is
$$ \rho_s^{\alpha}(t)= \frac{\text{Tr}_E\{(I\otimes P_\alpha)\rho_{SE}(t)(I\otimes P_\alpha)\}}{\text{Prob}(\alpha|\rho_s(t))} $$
I'm wondering how those two equations coming from? Also, since the indirect measurement aims to yield information about S without performing a projective (and thus destructive) direct measurement on S, why there's $P_\alpha$ in the equation? Thanks!!