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

Motivated by recent developments on calculus in metric measure spaces $(X,mathsf d,mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,mathsf d)$ and probability measures with barycenter in $L^q(X,mathfrak m)$, with $q$ dual exponent of $pin (1,infty)$. 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 (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)