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Viser: Measure, Integral and Probability
Measure, Integral and Probability
Marek Capinski og Peter E. Kopp
(2004)
Sprog: Engelsk
Springer
387,00 kr.
Print on demand. Leveringstid vil være ca 2-3 uger.
Detaljer om varen
- 2. Udgave
- Paperback: 311 sider
- Udgiver: Springer (Juli 2004)
- Forfattere: Marek Capinski og Peter E. Kopp
- ISBN: 9781852337810
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:· 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· 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 Fubini's theorem.-
6.5 Probability.-
6.6 Proofs of propositions.-
7. The Radon--Nikodym theorem.-
7.1 Densities and conditioning.-
7.2 The Radon--Nikodym theorem.-
7.3 Lebesgue--Stieltjes 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. 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 Fubini's theorem.-
6.5 Probability.-
6.6 Proofs of propositions.-
7. The Radon--Nikodym theorem.-
7.1 Densities and conditioning.-
7.2 The Radon--Nikodym theorem.-
7.3 Lebesgue--Stieltjes 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.
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