Author | Taylor, J. C. author |
---|---|
Title | An Introduction to Measure and Probability [electronic resource] / by J. C. Taylor |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1997 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0659-0 |
Descript | XVII, 324 p. 4 illus. online resource |
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. Kolmogorovโs inequality and the strong law of large numbers -- 5. Uniform integrability and truncation* -- 6. Differentiation: the HardyโLittlewood 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 GlivenkoโCantelli theorem -- 5. The characteristic function -- 6. Uniqueness and inversion of the characteristic function -- 7. The central limit theorem -- 8. Additional exercises* -- 9. Appendix*