Capacities and Hausdorff measures on metric spaces
  In this article, we show that in a $Q$-doubling space $(X,d,\mu),$ $Q>1,$ that supports a $Q$-Poincar\'e inequality and satisfies a chain condition, sets of $Q$-capacity zero have generalized Hausdorff $h$-measure zero for $h(t)=\log^{1-Q-\epsilon}(1/t).$ 
Nijjwal Karak and Pekka Koskela
math.FA