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We prove the existence of minimizers in the class of negative definitemeasures on compact subsets of momentum space in the homogeneous setting underseveral side conditions (constraints). The method is to employ Prohorov'stheorem. Given a minimizing sequence of negative definite measures, we showthat, under suitable side conditions, a unitarily equivalent subsequencethereof is bounded. By restricting attention to compact subsets, fromProhorov's theorem we deduce the existence of minimizers in the class ofnegative definite measures.

Existence of Minimizers for Causal Variational Principles on Compact Subsets of Momentum Space in the Homogeneous Setting