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Erschienen in:
2018 | OriginalPaper | Buchkapitel
7. Absolute Continuity of Measures and Functions. The Connection Between Derivative and Integral
verfasst von : Vladimir Kadets
Erschienen in: A Course in Functional Analysis and Measure Theory
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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 holds in the sense of Lebesgue’s integration.
$$ \int _\alpha ^\beta {f'(t)dt} = f(\beta ) - f(\alpha ) $$