  We prove the existence of minimizers in the class of negative definite
measures on compact subsets of momentum space in the homogeneous setting under
several side conditions (constraints). The method is to employ Prohorov's
theorem. Given a minimizing sequence of negative definite measures, we show
that, under suitable side conditions, a unitarily equivalent subsequence
thereof is bounded. By restricting attention to compact subsets, from
Prohorov's theorem we deduce the existence of minimizers in the class of
negative definite measures.
