  We show a norm convergence result for the Laplacian on a class of
post-critically finite fractals with arbitrary Borel regular probability
measure which can be approximated by a sequence of finite-dimensional graph
Laplacians with corresponding discrete probability measures. As a consequence
other functions of the Laplacians (heat operator, spectral projections etc.)
converge as well in operator norm. One also deduces convergence of the spectrum
and the eigenfunctions in energy norm.
