Suppose that $\{X_t : Ω → S := \mathbb{R}^d, t\in T\}$ is a stochastic process with independent increments and let $\mathcal{B}_t :=\mathcal{B}_t^X$ (natural filtration) for all $t\in T$. Show, for all $0 ≤ s < t$, that $(X_t − X_s )$ is independent of $\mathcal{B}_s^X$ and then use this to show $\{X_t\}_{t\in T}$ is a Markov process with transition kernels defined by $0 ≤ s ≤ t$, $$q_{s,t}(x, A) := E [1_A (x + X_t − X_s )]\text{ for all }A\in \mathcal{S}\text{ and }x\in\mathbb{R}^d.$$
The first part showing that $X_t-X_s$ is independent of $\mathcal{B}^X_s$. I more or less understand from a monotone class lemma.
For the part where I need to compute the transition kernel, I am not sure what I have to show. Is seems to me I have to show $P(X_t\in A|X_s)=q_{s,t}(X_s,A)$, is that correct? To do this I observe \begin{align} P(X_t\in A|X_s)=E(1_{X_t\in A}|X_s)=E(1_A(X_t)|X_s)=E(1_A(X_t-X_s+X_s)|X_s) \end{align} But then I am not sure how to finish. The exercise hintes that I should use that $X_t-X_s$ is independent of $\mathcal{B}^X_s$.