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

Luigi Ambrosio; Simone Di Marino; Giuseppe Savaré — 2015

Journal of the European Mathematical Society

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 $L^{q} (X, m)$ , 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...

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On the duality between p -modulus and probability measures

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