Fundamental Theorem of Calculus for Lebesgue Integration

The existing proofs of the Fundamental theorem of calculus for Lebesgue integration typically rely either on the Vitali–Carathéodory theorem on approximation of Lebesgue integrable functions by semi-continuous functions (as in [3, 9, 12]), or on the theorem characterizing increasing functions in terms of the four Dini derivates (as in [6, 10]). Alternatively, the theorem is derived using the Perron or the Kurzweil– Henstock integral and its relation to the Lebesgue integral (see [5, 8]). In this note we give a proof of the theorem which uses only standard results of the Lebesgue measure and integration without resorting to any extraneous material. Two of these results, the theorem that an absolutely continuous function with derivative equal to zero almost everywhere is constant, and Lebesgue’s theorem on differentiation of monotonic functions, have received an elegant treatment by elementary means in this Monthly in the hands of Michael Botsko [1, 2]. To simplify formulations we employ the following often used terminology. A statement is true nearly everywhere in S ⊂ R if it is true in S except for a countable subset of S. The idea for the proof of the following key lemma comes from [7].