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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...