  We define a filtration indexed by the integers on the tensor product of an
integrable highest weight module and a loop module for a quantum affine
algebra. We prove that the filtration is either trivial or strictly decreasing
and give sufficient conditions for this to happen. In the first case we prove
that the module is irreducible and in the second case we prove that the
intersection of all the modules is zero, thus allowing us to define the
completed tensor product. In certain special cases, we identify the subsequent
quotients of filtration. These are certain highest weight integrable modules
and the multiplicity and the highest weight are the same as that obtained by
decomposing the tensor product of the highest weight crystal bases with the
crystal bases of a loop module.
