Summary: Given a compact Riemann surface \(Y\) and a positive integer \(m\), M. S. Narasimhan and R. R. Simha [Invent. Math. 5, 120–128 (1968; Zbl 0159.37902)] 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 both of these measures to the singular curves on the boundary of \(\overline{\mathcal{M}_g}\) in such a way that they form a continuous family of measures on the universal curve over \(\overline{\mathcal{M}_g} \).