Integration with Respect to Finitely Additive Measures

This essay interprets the theory of finitely additive measures within the framework of the theory of Riesz spaces. The following topics are discussed: the extension procedures of measures, the Riemann and the Dunford integration procedures, the Radon-Nikodym Theorem and the Hahn Decomposition Theorem, the representation theory of the Radon- Nikodym derivatives as generalized functions, conditional expectation operators, the theory of L^p -spaces, and the norm completeness problem. The nature of the classical axiom of countable additivity is examined from Carathéodory’s algebraic measure-theoretic point of view