Phase space Feynman path integrals of higher order parabolic type with general functional as integrand

We give a general class of functionals for which the phase space Feynman path integrals of higher order parabolic type have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the endpoint of position paths and to the starting point of momentum paths. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the integration with respect to time, the interchange of the order with a limit and the perturbation expansion formula hold in the path integrals.