Suppose $X_t$ is a Lévy process, and denote its transition kernel $$ \kappa_{s,t}(x,A) = \mathbb{P}\left(X_t \in A \rvert X_s=x\right).$$ It is a known result that these transition kernels are homogeneous in both space and time, i.e $$ \kappa_{s,t}(x,A) = \kappa_{o,s-t}(0,A-x).$$
I tried proving the space homogeneity, for which I need to show that $$ \kappa_{s,t}(x,A) = \kappa_{s,t}(0,A-x).$$ I proceeded as follows:
We have that $$\kappa_{s,t}(x,A) = \mathbb{P}(X_t \in A \rvert X_s=x) = \mathbb{P}(X_t-X_s+x \in A \rvert X_s=x) = \mathbb{P}(X_t-X_s \in A-x \rvert X_s=x).$$ By the independent of increments (using $X_s=X_s-X_0$), we know that $X_t-X_s$ and $X_s$ are independent, and therefore $$ \kappa_{s,t}(x,A) = \mathbb{P}(X_t-X_s \in A-x \mid X_s=0) = \mathbb{P}(X_t \in A-x \mid X_s=0) = \kappa_{s,t}(0,A-x).$$
Is this correct? It feels fishy to 'fill in' the value $X_s=x$ in the probability for the set $A-X_s$ but not for the random variable $X_t-X_s$... Why is this allowed?
Thank you in advance! -Jonas