We consider a Markov process X(t) with extended generator A and domain D(A). Let F_t be a right-continuous history filtration and P_t denote the restriction of P to F_t. Let Q be another probability measure such that dQ_t/d P_t=E^h(t)$, where E^h(t) is a true exponential martingale depending on a positive function h from the domain D(A). We demonstrate that the process X(t) is a Markov process on the new probability space, we find its extended generator B and provide sufficient conditions under which D(B)=D(A). We apply this result to continuous time Markov chains, piecewise deterministic Markov processes and to diffusion processes (in this case a special choice of h yields the classical Cameron-Martin-Girsanov Theorem).