Markov Processes: $P_x$ and $E_x$

Question

In the study of Markov processes, one usually introduces the measures $P_{\pi}$ on the path space of the process where $\pi$ is an initial distribution of the process $X$ i.e $\pi=\mathcal L(X_0)$. The expectation w.r.t to this measure is denoted $E_{\pi}$. In the case $\pi=\delta_x$ one writes $P_x$ and $E_x$. Here and there I have seen remarks regarding the relationship between $P_x$ and $P(\cdot\mid X_0=x)$. It seems they should be versions of each other. Is this true, i.e $$P_x[X\in A]\overset{?}{\overset{\text{a.s}}{=}}P[X\in A\mid X_0=x]$$ If so, is it supposed to be obvious by definition, or can it be somehow formally derived? It seems that it's just "obvious by definition" by I don't want to be careless.

Answer 1

Yes, this is the definition (and a reason why the "a.s." above the equal sign is absurd).