A fine classification of Lebesgue points

Abstract Using Calderón–Zygmund techniques, we establish a link between gauge-controlled mean oscillation bounds and Hölder-type continuity. We introduce a gauge-based notion of Lebesgue set and examine its measurability, together with its approximation properties. For power gauges, we prove rectifiability results showing that, up to negligible sets, the graphs of functions on their Lebesgue sets can be covered by countably many Hölder graphs.