I am trying to understand what the follwing means
$$ v(x) = \sup_\tau \mathbb{E}_x(f(X_\tau)) = \sup_\tau \mathbb{E}(f(X_\tau) | X_0 = x) = \sup_\tau \frac{\mathbb{E}(1_{X_0 =x} f(X_\tau))}{P(X_0 = x)}$$
The first equation is the definition in the book (optimal stopping), what I am trying to understand. The second equation then is from the script. And the third is from wiki.
My problem is that $X$ is supposed to be a Levy process which means $P(X_0=0)=1$ which means to me that $P(X_0 = x) =0$. SO whats the deal with $x$ ?