Let $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued Lévy process. $X$ is a space- and time-homogeneous Markov process with transition semigroup $$\kappa_t(x,B):=\operatorname P\left[X_t\in B-x\right]\;\;\;\text{for }(x,B)\in E\times\mathcal B(E).$$
Let $$\pi(B):=|\{s\in(0,1]:0\ne\Delta X_s\in B\}\|\;\;\;\text{for }B\in\mathcal B(E).$$ Are we able to show that $$\lambda(B):=\operatorname E\left[\pi(B)\right]\;\;\;\text{for }B\in\mathcal B(E)$$ is a translation invariant measure?
I'm quite sure that this follows from the space-homgenity of $(\kappa_t)_{t\ge0}$ ($\kappa_t(x,B)=\kappa_t(0,B-x)$, but since not $\Delta X$ instead of $X$ is occurring in the definition of $\mu$, I'm not sure how we need to argue.