Dirichlet forms and potential theory of symmetric Hunt processes (Q2638670)

The authors provide a characterization of the class \(S_ 0\) of measures of ``finite energy integral'', in the context of a regular Dirichlet space (\({\mathcal F},{\mathcal E})\). [The class \(S_ 0\) features prominently in the monograph of \textit{M. Fukushima}, Dirichlet forms and Markov processes (1980; Zbl 0422.31007).] The characterization of \(S_ 0\) is then applied to study the invariant measures of the Markov process associated with (\({\mathcal F},{\mathcal E})\), and to give a new proof of the theorem of ``spectral synthesis''. [Reviewer's remark: The proof of Lemma 1.1 is invalid unless \(\nu R_{\alpha_ 0}\) is \(\sigma\)-finite; no difficulties result since \(\nu R_{\alpha_ 0}\) is \(\sigma\)-finite in all applications of the lemma.]