Absolute Continuity of Measures and Functions. The Connection Between Derivative and Integral

Abstract

In this chapter, after providing with necessary preliminaries, we present Hahn’s theorem on the positivity and negativity sets of a charge, the Radon–Nikodým theorem on absolutely continuous measures and give a complete description of functions $f:[a, b] \rightarrow \mathbb R$ for which on every $[\alpha , \beta ] \subset [a, b]$ the Newton–Leibniz formula

$$ \int _\alpha ^\beta {f'(t)dt} = f(\beta ) - f(\alpha ) $$

holds in the sense of Lebesgue’s integration.