Chapter 1. Riemann integration 5 §1.1. The Riemann integral: A review 5 §1.2. Characterization of Riemann integrable functions 18 §1.3. Historical notes: The integral from antiquity to Riemann 30 §1.4. Drawbacks of the Riemann integral 36 Chapter 2. Recipes for extending the Riemann integral 45 §2.1. A function theoretic view of the Riemann integral 45 §2.2. Lebesgue's recipe 47 §2.3. Riesz-Daniel recipe 49 Chapter 3. General extension theory 51 §3.1. First extension 51 §3.2. Semi-algebra and algebra of sets 54 vii
Let G be a locally compact Hausdorff abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f : G -*• X, with the topology of uniform convergence on compact sets. Let X' denote the dual of X with the weak* topology. Let M C (G, X') denote the space of all X'-valued compactly supported regular measures of finite variation on G.
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