Geodesic
Closability, regularity, and approximation by graphs for separable bilinear forms
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 299-317

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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. 
@article{ZNSL_2015_441_a18,
     author = {M. Hinz and A. Teplyaev},
     title = {Closability, regularity, and approximation by graphs for separable bilinear forms},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {299--317},
     publisher = {mathdoc},
     volume = {441},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a18/}
}
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M. Hinz; A. Teplyaev. Closability, regularity, and approximation by graphs for separable bilinear forms. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 299-317. http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a18/