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

Abstract

Motivated by recent developments on calculus in metric measure spaces $(X,\mathrm d,\mathrm m)$, we prove a general duality principle between Fuglede's notion \[15] of $p$-modulus for families of finite Borel measures in $(X,\mathrm d)$ and probability measures with barycenter in $L^q(X,\mathrm m)$, 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 $p$-podulus \[21, 23] and suitable probability measures in the space of curves (\[6, 7]).