On the duality between -modulus and probability measures
Luigi Ambrosio
Scuola Normale Superiore, Pisa, ItalySimone Di Marino
Scuola Normale Superiore, Pisa, ItalyGiuseppe Savaré
Università di Pavia, Italy
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Abstract
Motivated by recent developments on calculus in metric measure spaces , we prove a general duality principle between Fuglede's notion [15] of -modulus for families of finite Borel measures in and probability measures with barycenter in , with dual exponent of . We apply this general duality principle to study null sets for families of parametric and non-parametric curves in . 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 -podulus [21, 23] and suitable probability measures in the space of curves ([6, 7]).
Cite this article
Luigi Ambrosio, Simone Di Marino, Giuseppe Savaré, On the duality between -modulus and probability measures. J. Eur. Math. Soc. 17 (2015), no. 8, pp. 1817–1853
DOI 10.4171/JEMS/546