Limit of Weierstrass Measure on Stable Curves

The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $g\geqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne-Mumford compactification $\bar{\mathcal M}_g$ of the moduli $\mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $\mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.