Equality of the Capacity and the Modulus of a Condenser in Finsler Spaces

The equality of capacity and modulus is of great importance in the geometric theory of functions. It relates the functional-theoretic and geometric properties of sets. Special cases of the equality were proved by Ahlfors, Beurling, Fuglede, Ziemer, and Hesse. For the case of Euclidean metric, the equality of the capacity and the modulus was proved by Shlyk. The result of Aseev on the completeness of the system of continuous admissible functions was significantly used in Shlyk's proof. Finsler spaces were introduced as a generalization of Riemannian manifolds to the case in which the metric depends not only on the coordinates, but also on the direction. In the present paper, we prove the equality of the capacity and the modulus of a condenser in Finsler spaces under the most general assumptions. This result allows one to extend many results for Lebesgue function spaces to the case of function spaces with Finsler metric.