  The restricted Feynman path integrals (RFPIs) have been proposed to study
continuous quantum measurements in physics. The RFPIs are heuristically
determined in terms of the usual probability amplitude multiplied by weight for
each path, which contains information about the results and the resolution of
the measuring device. In the present paper we will consider the RFPIs
particularly for the position measurements and will prove rigorously that these
RFPIs are well defined in the $L^{2}$ space and are the solutions to the
non-self-adjoint Schroedinger equations. Our results in the present paper give
a generalization of the results on the usual Feynman path integrals for the
Schroedinger equations.Furthermore, our results are extended to quantum spin
systems.
