Real Analysis and Probability (Cambridge Studies in Advanced Mathematics)
Cambridge University Press
Edition: 2, 10/17/2002
EAN 9780521809726, ISBN10: 052180972X
Hardcover, 568 pages, 22.8 x 15.2 x 4 cm
Language: English
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.
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.