Measure, Integral and Probability

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Sprache: Englisch

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Beschreibung

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 modeling, discussed briefly from a measure-theoretical perspective to help the reader understand the underlying mathematical framework.

Zusammenfassung

Inhaltsverzeichnis

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.

Erscheinungsjahr:	2004
Fachbereich:	Wahrscheinlichkeitstheorie
Genre:	Importe , Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Reihe:	Springer Undergraduate Mathematics Series
Inhalt:	xv 311 S. 3 s/w Illustr. 311 p. 3 illus.
ISBN-13:	9781852337810
ISBN-10:	1852337818
Sprache:	Englisch
Einband:	Kartoniert / Broschiert
Autor:	Capinski, Marek Kopp, Peter E.
Auflage:	Second Edition 2004
Hersteller:	Springer Springer-Verlag London Ltd. Springer Undergraduate Mathematics Series
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	235 x 178 x 18 mm
Von/Mit:	Marek Capinski (u. a.)
Erscheinungsdatum:	27.08.2004
Gewicht:	0,573 kg