Levy Process is a Feller Processs

Question

I want to show that every Levy Process is a Feller Process. Let $ X=(X_t)_{t\geq 0}$ be a Levy Process and $\mu_t(dy):=P\circ X^{-1}(dy)$ . I found a proof where it is shown that the transition semigroup of a Levy Process \begin{equation} P_t\left(x,A\right)=\int_{\mathbb{R}^d} \mathbb{1}_A \left(x+y\right) \mu_t (dy) \end{equation} is a Feller semigroup.

How would I show that this is actually the transition semigroup of a given levy process, i.e. \begin{equation} \mathbb{E}\left[f\left(X_{s+t}\right)|\mathcal{F}_s\right]=P_t f\left(X_s\right) \end{equation} holds ?

Answer 1

We have

\begin{align} E[f(X_{t+s}-X_s +X_s) | \mathcal F_s]&=E[f(X_{t+s}-X_s +x)]\big|_{x = X_s}\\ &=E[f(X_t +x)]\big|_{x = X_s} \\ &=\int f(x+y)\mu_t(dy)\Big|_{x=X_s}\\ &=P_tf(X_s), \end{align} where the first equality comes from the so-called Freezing lemma (se e.g. here or wiki), second equality is due to $X_{t+s}-X_s$ has the same distribution as $X_t$, third equation is from the change of variable formula and the last one is from the definition of $P_t$.