  We start by showing how to approximate unitary and bounded self-adjoint
operators by operators in finite dimensional spaces. Using ultraproducts we
give a precise meaning for the approximation. In this process we see how the
spectral measure is obtained as an ultralimit of counting measures that arise
naturally from the finite dimensional approximations. Then we see how
generalized distributions can be interpreted in the ultraproduct. Finally we
study how one can calculate kernels of operators $K$ by calculating them in the
finite dimensional approximations and how one needs to interpret Dirac deltas
in the ultraproduct in order to get the kernels as propagators $\langle
x_{1}|K|x_{0}\rangle$.
