I'm reading a proof of the following proposition:

Let $(X_{t})_{t\geq 0}$ be a Lévy process on $\mathbb{R}^{d}$ and $B\subset\mathbb{R}^{d}.$ We deine $T_{B}=\inf\{t>0:X_{t}\in B\}$ and $T_{B}^{'}=\inf\{t\geq0:X_{t}\in B\}.$

Then $T_{B}^{'}$ is stopping time for $B$ open or closed set.

The proof given is the next: for the simple Markov property, $T_{B}^{\epsilon}=\inf\{t\geq\epsilon:X_{t}\in B\}$ is stopping time for all $\epsilon>0.$ The proof is followed.

I don't understand how Markov property works in this case. I know that $T_{B}$ is stopping time when $B$ is closed or open set.

If we assume that $T_{B}^{\epsilon}$ is stopping time, then $\displaystyle\lim_{\epsilon\rightarrow 0}T_{B}^{\epsilon}=T_{B}^{'},$ so we are done.

Any kind of help is thanked in advanced.


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