Marcinkiewicz integral and superdensity

Abstract In his beautiful book Singular Integrals and Differentiability Properties of Functions , E.M. Stein introduces Marcinkiewicz integral as an alternative to the differentiation theorem to illustrate (literally quoting from [1, Ch.I, Sect.2.3]) “the principle that a general point of a measurable set is almost completely surrounded by other points of the set”. We want to explore the theme summarised by this Stein’s sentence, using the notion of superdensity point.