Author	Capiลski, Marek. author
Title	Measure, Integral and Probability [electronic resource] / by Marek Capiลski, Peter Ekkehard Kopp
Imprint	London : Springer London, 2004
Edition	Second Edition
Connect to	http://dx.doi.org/10.1007/978-1-4471-0645-6
Descript	XV, 311 p. 3 illus. online resource

Measure, Integral and Probability is a gentle introduction that makes measure and integration theory accessible to the average third-year undergraduate student. The ideas are developed at an easy pace in a form that is suitable for self-study, with an emphasis on clear explanations and concrete examples rather than abstract theory. For this second edition, the text has been thoroughly revised and expanded. New features include: ยท a substantial new chapter, featuring a constructive proof of the Radon-Nikodym theorem, an analysis of the structure of Lebesgue-Stieltjes measures, the Hahn-Jordan decomposition, and a brief introduction to martingales ยท key aspects of financial modelling, including the Black-Scholes formula, discussed briefly from a measure-theoretical perspective to help the reader understand the underlying mathematical framework. In addition, further exercises and examples are provided to encourage the reader to become directly involved with the material

Content -- 1. Motivation and preliminaries -- 1.1 Notation and basic set theory -- 1.2 The Riemann integral: scope and limitations -- 1.3 Choosing numbers at random -- 2. Measure -- 2.1 Null sets -- 2.2 Outer measure -- 2.3 Lebesgue-measurable sets and Lebesgue measure -- 2.4 Basic properties of Lebesgue measure -- 2.5 Borel sets -- 2.6 Probability -- 2.7 Proofs of propositions -- 3. Measurable functions -- 3.1 The extended real line -- 3.2 Lebesgue-measurable functions -- 3.3 Examples -- 3.4 Properties -- 3.5 Probability -- 3.6 Proofs of propositions -- 4. Integral -- 4.1 Definition of the integral -- 4.2 Monotone convergence theorems -- 4.3 Integrable functions -- 4.4 The dominated convergence theorem -- 4.5 Relation to the Riemann integral -- 4.6 Approximation of measurable functions -- 4.7 Probability -- 4.8 Proofs of propositions -- 5. Spaces of integrable functions -- 5.1 The space L1 -- 5.2 The Hilbert space L2 -- 5.3 The LP spaces: completeness -- 5.4 Probability -- 5.5 Proofs of propositions -- 6. Product measures -- 6.1 Multi-dimensional Lebesgue measure -- 6.2 Product ?-fields -- 6.3 Construction of the product measure -- 6.4 Fubiniโs theorem -- 6.5 Probability -- 6.6 Proofs of propositions -- 7. The RadonโNikodym theorem -- 7.1 Densities and conditioning -- 7.2 The RadonโNikodym theorem -- 7.3 LebesgueโStieltjes measures -- 7.4 Probability -- 7.5 Proofs of propositions -- 8. LimitL theorems -- 8.1 Modes of convergence -- 8.2 Probability -- 8.3 Proofs of propositions -- Solutions -- References

1. Mathematics
2. Mathematical analysis
3. Analysis (Mathematics)
4. Measure theory
5. Economics
6. Mathematical
7. Probabilities
8. Mathematics
9. Probability Theory and Stochastic Processes
10. Analysis
11. Measure and Integration
12. Quantitative Finance