  Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan
and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the
canonical line bundle. We study the limit of this measure on holomorphic
families of Riemann surfaces with semistable reduction. The convergence takes
place on a hybrid space whose central fiber is the associated metrized curve
complex in the sense of Amini and Baker. We also study the limit of the measure
induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For
$m = 1$, both these measures coincide with the Bergman measure on $Y$. We also
extend the definition of the Narasimhan-Simha measure to the singular curves on
the boundary of $\overline{\mathcal{M}_g}$ in such a way that these measures
form a continuous family of measures on the universal curve over
$\overline{\mathcal{M}_g}$.
