  We provide a method for constructing (possibly non-trivial) measures on
non-locally compact Polish subspaces of infinite-dimensional separable Banach
spaces which, under suitable assumptions, are minimizers of causal variational
principles in the non-locally compact setting. Moreover, for non-trivial
minimizers the corresponding Euler-Lagrange equations are derived. The method
is to exhaust the underlying Banach space by finite-dimensional subspaces and
to prove existence of minimizers of the causal variational principle restricted
to these finite-dimensional subsets of the Polish space under suitable
assumptions on the Lagrangian. This gives rise to a corresponding sequence of
minimizers. Restricting the resulting sequence to countably many compact
subsets of the Polish space, by considering the resulting diagonal sequence we
are able to construct a regular measure on the Borel algebra over the whole
topological space.
  For continuous Lagrangians of bounded range it can be shown that, under
suitable assumptions, the obtained measure is a (possibly non-trivial)
minimizer under variations of compact support. Under additional assumptions, we
prove that the constructed measure is a minimizer under variations of finite
volume and solves the corresponding Euler-Lagrange equations. Afterwards, we
extend our results to continuous Lagrangians vanishing in entropy. Finally,
assuming that the obtained measure is locally finite, topological properties of
spacetime are worked out and a connection to dimension theory is established.
