# On the duality between $p$-modulus and probability measures

**submitted**
**year:** 2013

**abstract:** Motivated by recent developments on calculus in metric measure spaces $(X,\sfd,\mm)$, we prove a general
duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\sfd)$ and probability measures
with barycenter in $L^q(X,\mm)$, with $q$ dual exponent of $p\in (1,\infty)$. We apply this general duality principle to study null sets for families
of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation,
of the equivalence of notions of weak upper gradient based on $p$-Modulus and
suitable probability measures in the space of curves.

The paper is available on the
cvgmt preprint server.