Real Analysis and Probability
Real Analysis and Probability
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This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.
- Classic text by top-name author
- Comprehensive treatment makes it also a useful reference
- Over 400 exercises, many with hints for solutions
Reviews & endorsements
"A marvelous work which will soon become a standard text in the field for both teaching and reference...a complete and pedagogically perfect presentation of both the necessary preparatory material of real analysis and the proofs througout the text. Some of the topics and proofs are rarely found in other textbooks." Proceedings of the Edinburgh Mathematical Society
Product details
January 2005Adobe eBook Reader
9780511029585
0 pages
0kg
400 exercises
This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
- 1. Foundations: set theory
- 2. General topology
- 3. Measures
- 4. Integration
- 5. Lp spaces: introduction to functional analysis
- 6. Convex sets and duality of normed spaces
- 7. Measure, topology, and differentiation
- 8. Introduction to probability theory
- 9. Convergence of laws and central limit theorems
- 10. Conditional expectations and martingales
- 11. Convergence of laws on separable metric spaces
- 12. Stochastic processes
- 13. Measurability: Borel isomorphism and analytic sets
- Appendixes: A. Axiomatic set theory
- B. Complex numbers, vector spaces, and Taylor's theorem with remainder
- C. The problem of measure
- D. Rearranging sums of nonnegative terms
- E. Pathologies of compact nonmetric spaces
- Indices.
