  We consider the heat equation defined by a generalized measure theoretic
Laplacian on $[0,1]$. This equation describes heat diffusion in a bar such that
the mass distribution of the bar is given by a non-atomic Borel probabiliy
measure $\mu$, where we do not assume the existence of a strictly positive mass
density. We show that weak measure convergence implies convergence of the
corresponding generalized Laplacians in the strong resolvent sense. We prove
that strong semigroup convergence with respect to the uniform norm follows,
which implies uniform convergence of solutions to the corresponding heat
equations. This provides, for example, an interpretation for the mathematical
model of heat diffusion on a bar with gaps in that the solution to the
corresponding heat equation behaves approximately like the heat flow on a bar
with sufficiently small mass on these gaps.
