This classic graduate textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The comprehensive historical notes have been further amplified for this edition, and a number of new exercises have been added, together with hints for solution.
This classic graduate textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The comprehensive historical notes have been further amplified for this edition, and 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.
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.
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