  We consider a countably generated and uniformly closed algebra of bounded
functions. We assume that there is a lower semicontinuous, with respect to the
supremum norm, quadratic form and that normal contractions operate in a certain
sense. Then we prove that a subspace of the effective domain of the quadratic
form is naturally isomorphic to a core of a regular Dirichlet form on a locally
compact separable metric space. We also show that any Dirichlet form on a
countably generated measure space can be approximated by essentially discrete
Dirichlet forms, i.e. energy forms on finite weighted graphs, in the sense of
Mosco convergence, i.e. strong resolvent convergence.
