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 [15] of p-modulus for families of finite Borel measures in (X, d) and probability measures with barycenter in Lq(X,m), with q dual exponent of p ∈ (1,∞). 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 ppodulus ([21], [23]) and suitable probability measures in the space of curves ([6], [7]).