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2016 Chapter VII. Differentiation of Lebesgue Integrals on the Line

Abstract

This chapter concerns the Fundamental Theorem of Calculus for the Lebesgue integral, viewed from Lebesgue's perspective but slightly updated.

Section 1 contains Lebesgue's main tool, a theorem saying that monotone functions on the line are differentiable almost everywhere. A relatively easy consequence is Fubini's theorem that an absolutely convergent series of monotone increasing functions may be differentiated term by term. The result that the indefinite integral $\int_a^xf(t)\,dt$ of a locally integrable function $f$ is differentiable almost everywhere with derivative $f$ follows readily.

Section 2 addresses the converse question of what functions $F$ have the property for a particular $f$ that the integral $\int_a^bf(t)\,dt$ can be evaluated as $F(b)-F(a)$ for all $a$ and $b$. The development involves a decomposition theorem for monotone increasing functions and a corresponding decomposition theorem for Stieltjes measures. The answer to the converse question when $f\geq0$ and $F'=f$ almost everywhere is that $F$ is “absolutely continuous” in a sense defined in the section.

Information

Published: 1 January 2016
First available in Project Euclid: 26 July 2018

Digital Object Identifier: 10.3792/euclid/9781429799997-7

Rights: Copyright © 2016, Anthony W. Knapp

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