Measure, Integration & Real Analysis, 2021a

Chapter 9
Real and Complex Measures
A measure is a countably additive function from a σ-algebra to [0, ∞]. In this chapter,
we consider countably additive functions from a σ-algebra to either R or C. The first
section of this chapter shows that these functions, called real measures or complex
measures, form an interesting Banach space with an appropriate norm.
The second section of this chapter focuses on decomposition theorems that help
us understand real and complex measures. These results will lead to a proof that the
dual space of L p (μ) can be identified with L p′ (μ).
Dome in the main building of the University of Vienna, where Johann Radon
(1887–1956) was a student and then later a faculty member. The Radon–Nikodym
Theorem, which will be proved in this chapter using Hilbert space techniques,
provides information analogous to differentiation for measures.
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Measure, Integration & Real Analysis, by Sheldon Axler
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