Inserted: 6 nov 2013
Last Updated: 29 aug 2014
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.