Lebesgue points and capacities via the boxing inequality in metric spaces

Abstract. The purpose of this work is to study regularity of So-bolev functions on metric measure spaces equipped with a doub-ling measure and supporting a weak Poincare ́ inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codi-mension one and study characterizations of 1-capacity in terms of Frostman’s lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin’s boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperi-metric inequalities and coarea formula play an essential role in our approach. 1