Properties of capacities generated by Sobolev classes on metric measure spaces

Properties of $\mathrm{Cap}_{\alpha, p}$–capacities, generated by Sobolev classes on metric spaces with doubling measure, are investigated. The principal case is $p>0$ not investigated before. It's proved that capacity is an outer measure; the property of continuity, the relation between different capacities, measure and Hausdorff dimension is investigated.