The main result reads as follows. Let \(R \leq \infty\) and \(F_{R}^{\epsilon}\) and \(F_{R}\) be the energy functionals defined in \(L^2(\Omega_R, d \mu^\epsilon)\) and \(L^2(\Omega_R, d \mu^\prime)\), respectively. It follows that \(F_{R}^{\epsilon}\) and \(F_{R}\) are local and regular Dirichlet forms. Assume \(R 0, \] and the associated spectral measures \(E^\epsilon\) and \(E\) satisfy \[ E^{\epsilon}((\lambda,\eta]) \rightarrow E((\lambda,\eta])\text{ as }\epsilon \to 0 \] for every \(\lambda 0\) and \(\epsilon(m) > 0\) be sequences tending to 0 as \(m \rightarrow \infty\) and let \(u_m \in L^2(\Omega_R, d \mu^{\epsilon(m)})\) satisfy: \[ F_R^{\epsilon(m)}[u_m] 0}\) associated to \(F^\epsilon\) is tight. Furthermore, if \(0\leq \beta, \gamma, \nu \leq 1\) and \(\alpha\geq 2 \max \{\beta, \gamma\}\), then \(\{{\mathbb P}^\epsilon\}\) weakly converges to the Wiener measure \({\mathbb P}\) associated to \(F\). In particular, \(F^\epsilon\) converges to \(F\) in both Mosco and \(\Gamma\) senses.