  In this paper, we prove that the L^2 Betti numbers of an amenable covering
space can be approximated by the average Betti numbers of a regular exhaustion,
under some hypotheses. We also prove that some L^2 spectral invariants can be
approximated by the corresponding average spectral invariants of a regular
exhaustion. The main tool which is used is a generalisation of the "principle
of not feeling the boundary" (due to M. Kac), for heat kernels associated to
boundary value problems.
