| 这是一本广受称赞的教科书，清晰地讲解了现代概率论以及度量空间与概率测度之间的相互作用。本书在两个方面获得了极佳的成功。一是它是一本全面、新颖的实分析教程，二是它是一本数学理论完整和自成体系的概率论教程。 |
| R. M. Dudley，麻省理工学院数学教授。除本书外，他还著有《Differentiability of Six Operators on Nonsmooth Functions and p-Variation》、《Uniform Central Limit Theorems》等书。... .. << 查看详细 |
| preface to the cambridge edition . 1 foundations; set theory 1.1 definitions for set theory and the real number systen 1.2 relations and orderings * 1.3 transfinite induction and recursion 1.4 cardinality 1.5 the axiom of choice and its equivalents 2 general topology 2.1 topologies, metrics, and continuity 2.2 compactness and product topologies 2.3 complete and compact metric spaces 2.4 some metrics for function spaces 2.5 completion and completeness of metric spaces *2.6 extension of continuous functions *2.7 uniformities and uniform spaces *2.8 compactification 3 measures 3.1 introduction to measures 3.2 semirings and rings 3.3 completion of measures .3.4 lebesgue measure and nonmeasurable sets *3.5 atomic and nonatomic measures 4 integration 4.1 simple functions *4.2 measurability 4.3 convergence theorems for integrals 4.4 product measures *4.5 daniell-stone integrals 5 lp spaces; introduction to functional analysis 5.1 inequalities for integrals 5.2 norms and completeness oflp 5.3 hilbert spaces 5.4 orthonormal sets and bases 5.5 linear forms on hilbert spaces, inclusions oflp spaces, and relations between two measures 5.6 signed measures 6 convex sets and duality of normed spaces 6.1 lipschitz, continuous, and bounded functionals 6.2 convex sets and their separation 6.3 convex functions *6.4 duality of lp spaces 6.5 uniform boundedness and closed graphs *6.6 the brunn-minkowski inequality 7 measure, topology, and differentiation 7.1 bake and borel a-algebras and regularity of measures *7.2 lebesgue's differentiation theorems *7.3 the regularity extension *7.4 the dual of c(k) and fourier series *7.5 almost uniform convergence and lusin's theorem 8 introduction to probability theory 8.1 basic definitions 8.2 infinite products of probability spaces .. 8.3 laws of large numbers *8.4 ergodic theorems 9 convergence of laws and central limit theorems 9.1 distribution functions and densities 9.2 convergence of random variables 9.3 convergence of laws 9.4 characteristic functions 9.5 uniqueness of characteristic functions and a central limit theorem 9.6 triangular arrays and lindeberg's theorem 9.7 sums of independent real random variables *9.8 the l6vy continuity theorem; infinitely divisible and stable laws 10 conditional expectations and martingales 10.1 conditional expectations 10.2 regular conditional probabilities and jensen's inequality 10.3 martingales 10.4 optional stopping and uniform integrability 10.5 convergence of martingales and submartingales * 10.6 reversed martingales and submartingales * 10.7 subadditive and superadditive ergodic theorems 11 convergence of laws on separable metric spaces 11.1 laws and their convergence 11.2 lipschitz functions 11.3 metrics for convergence of laws 11.4 convergence of empirical measures 11.5 tightness and uniform tightness * 11.6 strassen's theorem: nearby variables with nearby laws * 11.7 a uniformity for laws and almost surely converging realizations of converging laws * 11.8 kantorovich-rubinstein theorems * 11.9 u-statistics 12 stochastic processes 12.1 existence of processes and brownian motion 12.2 the strong markov property of brownian motion 12.3 reflection principles, the brownian bridge, and laws of suprema 12.4 laws of brownian motion at markov times: skorohod imbedding 12.5 laws of the iterated logarithm 13 measurability: borel isomorphism and analytic sets * 13.1 borel isomorphism 13.2 analytic sets appendix a axiomatic set theory a.1 mathematical logic a.2 axioms for set theory a.3 ordinals and cardinals a.4 from sets to numbers appendix b complex numbers, vector spaces, and taylor's theorem with remainder appendix c the problem of measure appendix d rearranging sums of nonnegative terms appendix e pathologies of compact nonmetric spaces author index subject index notation index ... |
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