Lebesgue points and capacities via the boxing inequality in metric spaces

The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincaré 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 codimension 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, isoperimetric inequalities and coarea formula play an essential role in our approach. Indiana University Mathematics Journal