Fine densities for excessive measures and the Revuz correspondence
Let \({\mathcal U}=(U_\alpha)_{\alpha>0}\) be a proper resolvent of a Borel right Markov process on \(X\) and \(\xi=\rho\circ U+h\) a \({\mathcal U}\)-excessive measure, where \(\rho\circ U\) and \(h\) are the potential and the harmonic parts, respectively. A nearly Borel function \(f\) is called \(\xi\)-fine if there exists a set \(N\) which is \(\xi\)-polar and \(\rho\)-negligible such that \(f\) is finely continuous on \(X\setminus N\). It is shown by an analytic method that every \({\mathcal U}\)-excessive measure \(\eta\) such that \(\eta \ll \xi\) has a \(\xi\)-fine version of \(d\eta/d\xi\). For a strongly supermedian kernel \(V\), the Revuz measure of \(V\) is defined by \(\nu^\xi_V(B)=L(\xi,V(I_B))\), where \(L\) is the energy functional. Then the above result is used to give the Revuz correspondence between the regular (strongly) supermedian kernels and the measures charging no \(\xi\)-semipolar (polar) sets which are \(\rho\)-negligible. This result is applied to the semi-Dirichlet form \(({\mathcal E}, {\mathcal D}({\mathcal E}))\) on \((X,m)\) to get the correspondence between the measure \(\mu\) charging no \(m\)-polar sets and regular excessive kernel \(V\) by the relation \(\mu(f \tilde{g})={\mathcal E}_\alpha(V^\alpha f,g)\), where \(\tilde{g}\) is a quasi-continuous modification of \(g\).
scientific article; zbMATH DE number 2057494