  Consider a BV function on a Riemannian manifold. What is its differential?
And what about the Hessian of a convex function? These questions have clear
answers in terms of (co)vector/matrix valued measures if the manifold is the
Euclidean space. In more general curved contexts, the same objects can be
perfectly understood via charts. However, charts are often unavailable in the
less regular setting of metric geometry, where still the questions make sense.
  In this paper we propose a way to deal with this sort of problems and, more
generally, to give a meaning to a concept of `measure acting in duality with
sections of a given bundle', loosely speaking. Despite the generality, several
classical results in measure theory like Riesz's and Alexandrov's theorems have
a natural counterpart in this setting. Moreover, as we are going to discuss,
the notions introduced here provide a unified framework for several key
concepts in nonsmooth analysis that have been introduced more than two decades
ago, such as: Ambrosio-Kirchheim's metric currents, Cheeger's Sobolev functions
and Miranda's BV functions.
  Not surprisingly, the understanding of the structure of these objects
improves with the regularity of the underlying space. We are particularly
interested in the case of $\RCD$ spaces where, as we will argue, the regularity
of several key measures of the type we study nicely matches the known
regularity theory for vector fields, resulting in a very effective theory.
  We expect that the notions developed here will help creating stronger links
between differential calculus in Alexandrov spaces (based on Perelman's DC
charts) and in $\RCD$ ones (based on intrinsic tensor calculus).
