I have problems to understand the strong Markov property (Klenke, p. 356):
Let $I \subset [0,\infty)$ be closed under addition. A Markov process $(X_t)_{t\in I}$ with distributions $(\mathbf{P}_x, x \in E)$ has the strong Markov property if, for every a.s. finite stopping time $\tau$, every bounded $\mathcal{B}(E)^{\otimes I}-\mathcal{B}(\mathbb{R})$ measurable function $f : E^I \rightarrow \mathbb{R}$ and every $x \in E$, we have $$\mathbf{E}_x\Bigl[f\bigl((X_{\tau+t})_{t\in I}\bigr) \big| \mathcal{F}_\tau \Bigr] = \mathbf{E}_{X_\tau}\Bigl[f(X)\Bigr] := \int_{E^I} \kappa(X_\tau, dy) f(y)\, .$$
If $E \subset \mathbb{R}$ can this formulation be simplified?
And in general, is there an easier formulation somewhere? Thank you!