The Dirichlet Laplacian on Finely Open Sets

For any decreasing sequence of bounded finely open sets D_i R^N it is shown that, for every n, the nth eigenvalue _n ( D_i) of the Dirichlet laplacian A ( D_i ) on D_i converges to _n ( D ) (the nth eigenvalue of A ( D ) ), where D denotes the fine interior of D_i. Likewise, A ( D_i )^-1 A ( D )^-1 in operator norm. Similar results are obtained for increasing or just order convergent sequences ( D_i ). Furthermore, A ( D )^-1 is identified with the integral operator on L² ( D ) whose kernel is Green's function for D.