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Given a Markov family $X=\{X_t, \mathcal{F}_t, t\ge 0\}$ on some $(\Omega,\mathcal{F})$,together with a family of probability measures $\{P^x\}_{x \in \mathbb{R}^d}$ on $(\Omega,\mathcal{F})$ and a nice function $f$ so that $f(X_t)$ is always integrable w.r.t to the entire spectrum of probability measures $\{P^x\}_{x \in \mathbb{R}^d}$, and S is a finite optional time. How do I interpret the following function?

$E^{X_S}[f(X_t)]$

Can I think of think of it as $\omega \mapsto E^{X_{S(\omega)}(\omega)}[f(X_t)]$ This interpretation seems very strange to me since we first fix $\omega$ to compute $X_S(\omega)(\omega)=x$ and then compute the expectation of the random variable $\omega \mapsto X_t(\omega)$ on $\Omega$ w.r.t to the probability measure $P^x$.

This is bothering me since I cant follow statement for example in $e'$ in Proposition 6.7 shown below in Karatzas and Shreve enter image description here

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    $\begingroup$ Note that $P_t f(x) := \mathbb{E}^x f(X_t)$ is the semigroup associated with the Markov process $(X_t)_{t \geq 0}$. The expression $\mathbb{E}^{X_s} f(X_t)$ hence equals $P_t f(X_s)$. Nothing too strange about that.... $\endgroup$ – saz Dec 3 '18 at 15:48

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