Causal variational principles ask for a measure on a topological space that minimizes a double integral of a symmetric, non-negative Lagrangian under a volume constraint. When the underlying space is an infinite-dimensional, non-locally compact Polish subspace of a separable Banach space, the classical compactness arguments fail. This concept covers the constructive strategy that recovers existence anyway: exhaust the Banach space by an increasing chain of finite-dimensional subspaces where the problem is locally compact, obtain a minimizer on each, then use tightness, Prohorov's theorem and a Cantor diagonal argument over a countable family of compact sets to assemble a regular global Borel measure, and identify the extra conditions under which that measure genuinely minimizes under variations of compact support and of finite volume and satisfies the corresponding Euler-Lagrange equations.
This paper presents a constructive method for establishing existence of minimizers for causal variational principles on non-locally compact Polish subspaces of infinite-dimensional separable Banach s…