The Lebesgue differentiation theorem via the rising sun lemma
An elegant proof of Lebesgue's differentiation theorem is obtained using a version of the rising sun lemma for continuous functions and a reduction of the main statement to the continuous case, based on the fact that a strictly increasing function on an interval has a continuous inverse. Only basic properties of Lebesgue outer measure are needed, making this a suitable approach for an elementary course.
scientific article; zbMATH DE number 2151959