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.