  We develop an asymptotic expansion of the spectral measures on a degenerating
family of hyperbolic Riemann surfaces of finite volume. As an application of
our results, we study the asymptotic behavior of weighted counting functions,
which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T)
= \sum\limits_{\lambda_n \leq T}(T-\lambda_n)^w $$ where $\{\lambda_n\}$ is the
set of eigenvalues of the Laplacian which acts on the space of smooth functions
on $M$. If $M$ is non-compact, then the weighted counting function is defined
via the inverse Laplace transform. Now let $M_{\ell}$ denote a degenerating
family of compact or non-compact hyperbolic Riemann surfaces of finite volume
which converges to the non-compact hyperbolic surface $M_{0}$. As an example of
our results, we have the following theorem: There is an explicitly defined
function $G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such that
for $w > 3/2$ and $T>0$, we have $$N_{M_{\ell},w}(T) = G_{\ell,w}(T)
+N_{M_{0},w}(T) +o(1)$$ for $\ell \to 0$. We also consider the setting when $w
< 3/2$, and we obtain a new proof of the continuity of small eigenvalues on
degenerating hyperbolic Riemann surfaces of finite volume.
