The Lebesgue differentiation theorem revisited

Abstract

We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involves the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization of the Lebesgue measurable functions on R n in terms of approximate continuity associated to an expansive linear map.

Perbandingan Penyelesaian Integral Riemann, Lebesgue dan HK Berdasarkan Definisi

“…Dalam perkembangannya ditemukan fungsi-fungsi takterintegralkan Riemann, berikut kekurangan integral Riemann [8]: Pada tahun 1902 Lebesgue melakukan perumuman untuk mengatasi tiga kekurangan pada integral Riemann tersebut [8]. Hingga saat ini, beberapa literatur masih melakukan kajian tentang integral Lebesgue seperti [9], [10], [11], [12]. Meskipun dapat mengatasi kekurangan integral Riemann, dalam kajian sifat-sifat lanjutan integral Lebesgue masih terdapat fungsi 𝐹𝐹 kontinu di Riemann, dan integral Lebesgue [8] [13].…”