  The present paper is the first in a series devoted to the study of asymptotic
geometry of Riemann surfaces and their moduli spaces.
  We introduce the moduli space of hybrid curves as a new compactification of
the moduli space of curves, refining the one obtained by Deligne and Mumford.
This is the moduli space for multiscale geometric objects which mix complex and
higher rank tropical and non-Archimedean geometries, reflecting both discrete
and continuous features.
  We define canonical measures on hybrid curves which combine and generalize
Arakelov-Bergman measures on Riemann surfaces and Zhang measures on metric
graphs.
  We then show that the universal family of canonically measured hybrid curves
over this moduli space varies continuously. This provides a precise link
between the non-Archimedean Zhang measure and variations of Arakelov-Bergman
measures in families of Riemann surfaces, answering a question which has been
open since the pioneering work of Zhang on admissible pairing in the nineties.
