A rigorous real time Feynman path integral

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. (C) 1999 American Institute of Physics. [S0022-2488(99)03801-3].