On the duality between p-modulus and probability measures

Motivated by recent developments on calculus in metric measure spaces (X, d, m), we prove a general duality principle between Fuglede's notion of p-modulus for families of finite Borel measures in (X, d) and probability measures with barycenter in L-q (X, m), with q the dual exponent of p is an element of (1, infinity). We apply this general duality principle to study null sets for families of parametric and nonparametric 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.