probability theory - Possibly broken definition of the strong Markov property

Let

• $I\subseteq [0,\infty)$ be closed under addition and $0\in E$


• $E$ be a Polish space and $\mathcal E$ be the Borel $\sigma$-algebra on $E$


• $X=(X_t)_{t\in I}$ be a Markov process with values in $(E,\mathcal E)$ and distributions $(\operatorname P_x)_{x\in E}$


• $\mathbb F=(\mathcal F_t)_{t\in I}$ be the filtration generated by $X$

$X$ is said to have the strong Markov property $:\Leftrightarrow$ For all almost surely finite $\mathbb F$-stopping times $\tau$, $x\in E$ and bounded, $\mathcal E^{\otimes I}$-measurable $f:E^I\to\mathbb R$, $$\operatorname E_x\left[f\left(\left(X_{\tau+t}\right)_{t\in I}\right)\mid\mathcal F_\tau\right]=\operatorname E_{X_\tau}\left[f\left(X\right)\right]\;\;\;\operatorname P_x\text{-almost surely}\;,\tag 1$$ where $\mathcal F_\tau:=\left\{A\in\mathcal A:A\cap\left\{\tau\le t\right\}\in\mathcal F_t\;\text{for all }t\in I\right\}$.

I'm curious about two things:

1. What's the reason to force $\tau$ to be almost surely finite? What's meant by almost surely at all (with respect to which probability measure?), in this context?


2. Unless $\tau$ is $\operatorname P_x$-almost surely finite, the integrand on the left and the expression on the right side of $(1)$ seem to undefined on $\left\{\tau=\infty\right\}$

So, is the given definition broken? If that's the case: What do we need to change to fix it?