This is part of a statement from From Levy type processes to Parabolic SPDEs by Rene SChilling.
Corollary 2.4 states that the finite dimensional distributions $P(X_{t_1}\in dx_1 ,\dots, X_{t_n}\in dx_n)$ of a Levy process are uniquely determined by $$E \exp (i\sum_{k=1}^n \xi_k \cdot X_{t_k})=\Pi_{i=1}^n [E\exp(i(\xi_k + \cdots + \xi_n)\cdot X_1)]^{t_k - t_{k-1}}.$$Then it says it is not hard to invert the above Fourier transform and writing $p_t(dx):=P(X_t \in dx)$ we get $$P(X_{t_1}\in B_1 , \dots , X_{t_n} \in B_n)=\int \cdots \int \Pi_{k=1}^n 1_{B_k}(x_1 + \cdots + x_k)p_{t_k - t_{k-1}}(dx_k).$$
I am having a hard time figuring out how to get this integral form of the joint distribution from the characteristic function. I would greatly appreciate some help.