Many students in mathematics, statistics, finance, business, and engineering need an introduction to measure theory. This book provides a self-contained introduction that provides students with the mathematical background needed to study applications in their areas.
Auflage
1st ed. 1997. Corr. 2nd printing 1998
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Graduate
Editions-Typ
Illustrationen
4
4 s/w Abbildungen
XVII, 324 p. 4 illus.
Maße
Höhe: 235 mm
Breite: 155 mm
Dicke: 18 mm
Gewicht
ISBN-13
978-0-387-94830-0 (9780387948300)
DOI
10.1007/978-1-4612-0659-0
Schweitzer Klassifikation
I. Probability Spaces.- 1. Introduction to ?.- 2. What is a probability space? Motivation.- 3. Definition of a probability space.- 4. Construction of a probability from a distribution function.- 5. Additional exercises*.- II. Integration.- 1. Integration on a probability space.- 2. Lebesgue measure on ? and Lebesgue integration.- 3. The Riemann integral and the Lebesgue integral.- 4. Probability density functions.- 5. Infinite series again.- 6. Differentiation under the integral sign.- 7. Signed measures and the Radon-Nikodym theorem*.- 8. Signed measures on ? and functions of bounded variation*.- 9. Additional exercises*.- III. Independence and Product Measures.- 1. Random vectors and Borel sets in ?n.- 2. Independence.- 3. Product measures.- 4. Infinite products.- 5. Some remarks on Markov chains*.- 6. Additional exercises*.- IV. Convergence of Random Variables and Measurable Functions.- 1. Norms for random variables and measurable functions.- 2. Continuous functions and Lp*.- 3. Pointwise convergence and convergence in measure or probability.- 4. Kolmogorov's inequality and the strong law of large numbers.- 5. Uniform integrability and truncation*.- 6. Differentiation: the Hardy-Littlewood maximal function*.- 7. Additional exercises*.- V. Conditional Expectation and an Introduction to Martingales.- 1. Conditional expectation and Hilbert space.- 2. Conditional expectation.- 3. Sufficient statistics*.- 4. Martingales.- 5. An introduction to martingale convergence.- 6. The three-series theorem and the Doob decomposition.- 7. The martingale convergence theorem.- VI. An Introduction to Weak Convergence.- 1. Motivation: empirical distributions.- 2. Weak convergence of probabilities: equivalent formulations.- 3. Weak convergence of random variables.- 4.Empirical distributions again: the Glivenko-Cantelli theorem.- 5. The characteristic function.- 6. Uniqueness and inversion of the characteristic function.- 7. The central limit theorem.- 8. Additional exercises*.- 9. Appendix*.
