and $P_a(t)P_{a'}(t)= 0$ if the intervals surrounding $a$ and $a'$ do not have overlap.

We would like now to ask the question “what is the probability that the observable $O$ take a given value $O_0$ given that the clock indicates a certain time $T_0$ ”. Such question is embodied in the conditional probability, $$\label{condprob}
{\cal P}\left( O\in [O_0-\Delta O,O_0+\Delta O]|T \in 
[T_0-\Delta T,T_0+\Delta T]\right) =\lim_{\tau\to\infty}
{\int_{-\tau}^{\tau} dt\, {\rm Tr}\left(P_O(t) P_T(t)\rho P_T(t)\right) 
\over 
{\int_{-\tau}^{\tau} dt\,{\rm Tr}\left(P_T(t)\rho \right)}}$$ where we have used the properties of both projectors and the integrals over $t$ in the right hand side are taken over all its possible values. The reason for the integrals is that we do not know for what value of the external ideal time $t$ the clock will take the value $T_0$. In this expression $\rho$ is the density matrix of the system. One has to take some obvious cares, like for instance to choose a clock variable (or set of variables) that do not take twice the same value during the relevant lifetime of the experiment one is considering.

The above expression is general, it will apply to any choice of “clock” and “system” variables we make. The relational evolution of the 