  Using improper Riemann integrals, we will formulate a rigorous version of the
real-time, time-sliced Feynman path integral for the $L^2$ transition
probability amplitude. We will do this for nonvector potential Hamiltonians
with potential which has at most a finite number of discontinuities and
singularities. We will also provide a Nonstandard Analysis version of our
formulation.
