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I encountered a problem while having a look at the proof of the strong Markov Property in Liggett's book about Markov Processes "Continuous Time Markov Processes: An Introduction". I want to proof the Strong Markov Property for continuous Markov chains using the same approach as in the book. As far as I can see the proof in the book uses the following statement:

Let $(X_t)_{t\geq 0}$ be a continuous Markov chain with values in the discrete space $S$. Further let $Y$ be a bounded random variable and $\xi$ a finite stopping time. Then the following holds $Pr_x$-a.s.

$$ E_x[Y] = E_{x}[E_{\xi}[Y]] \iff E_x[E_x[Y\cdot \theta_{\xi} \vert {\cal{F}}_{\xi} ]] = E_{x}[E_{\xi}[Y]]$$

$$ \implies E_x[Y\cdot \theta_{\xi} \vert {\cal{F}}_{\xi} ] = E_{\xi}[Y] $$

where ${\cal{F}}_{\xi} := \{ A \in {\cal{F}}: A \cap \{\xi\leq t\}\in {\cal{F}}_t \text{ for all } t\geq 0 \}$ and $\theta_s$ the shift operator for right-continuous functions $\omega$ i.e. $(\theta_s \omega)(t) = \omega (t+s)$.

I have a problem with the implication in the second line of the equations. In general this doesn't hold because I can find two different random variables s.t. the expectation would be the same, right? So am I missing another argument that is being used to proof the statement?

Thank you for your help!

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For those who are running into the same problem, I figured out why the above holds. By proving that the expected values are equal, we show the definition of the conditioned expectation (see e.g. Durrett). The last step is to show that the random variable is indeed measurable.

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