Capacity and Hausdorff measure in Musielak-Orlicz-Sobolev spaces

In this paper, we show that sets with zero Sobolev $p(\cdot )$-capacity have generalized Hausdorff $h(\cdot )$-measure zero, for some gauge function $h(\cdot )\mathrm{.}$ We also prove that sets with zero Musielak-Orlicz-Sobolev $\mathrm{\Phi }(\cdot ,\cdot )$-capacity, for a particular class of functions $\mathrm{\Phi }(\cdot ,\cdot ),$ have generalized Hausdorff $h(\cdot )$-measure zero, for a suitable gauge function $h(\cdot )\mathrm{.}$