# Measure Theory and Probability Theory

Springer Science & Business Media, Jul 27, 2006 - Business & Economics - 618 pages

This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix.

The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement.

Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales.

Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes.

Krishna B. Athreya is a professor at the departments of mathematics and statistics and a Distinguished Professor in the College of Liberal Arts and Sciences at the Iowa State University. He has been a faculty member at University of Wisconsin, Madison; Indian Institute of Science, Bangalore; Cornell University; and has held visiting appointments in Scandinavia and Australia. He is a fellow of the Institute of Mathematical Statistics USA; a fellow of the Indian Academy of Sciences, Bangalore; an elected member of the International Statistical Institute; and serves on the editorial board of several journals in probability and statistics. Soumendra N. Lahiri is a professor at the department of statistics at the Iowa State University. He is a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association, and an elected member of the International Statistical Institute.

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### Contents

 III 9 IV 14 V 19 VII 25 VIII 27 IX 28 X 30 XI 31
 XCII 389 XCIII 392 XCIV 393 XCV 399 XCVI 405 XCVII 417 XCVIII 424 XCIX 425

 XII 39 XIII 44 XIV 47 XV 48 XVI 59 XVII 61 XVIII 71 XIX 83 XX 89 XXII 93 XXIII 94 XXIV 96 XXV 97 XXVI 98 XXVII 102 XXVIII 112 XXIX 119 XXX 125 XXXI 128 XXXII 133 XXXIII 134 XXXIV 136 XXXV 137 XXXVI 147 XXXVII 152 XXXVIII 157 XXXIX 160 XL 162 XLII 164 XLIV 166 XLV 173 XLVI 178 XLVII 181 XLVIII 189 XLIX 191 L 199 LI 212 LII 219 LIII 222 LIV 227 LV 236 LVI 240 LVII 249 LVIII 254 LIX 260 LX 262 LXI 264 LXII 266 LXIII 268 LXIV 271 LXV 274 LXVI 278 LXVII 279 LXVIII 287 LXIX 291 LXX 299 LXXI 303 LXXII 306 LXXIII 307 LXXV 309 LXXVI 316 LXXVII 323 LXXVIII 327 LXXIX 332 LXXX 337 LXXXI 343 LXXXII 352 LXXXIII 358 LXXXIV 361 LXXXVI 364 LXXXVII 368 LXXXVIII 372 LXXXIX 374 XC 376 XCI 383
 CI 427 CII 429 CIII 430 CV 438 CVI 440 CVII 442 CVIII 443 CIX 457 CX 458 CXI 461 CXII 462 CXIII 464 CXIV 465 CXV 467 CXVI 469 CXVII 473 CXVIII 477 CXIX 478 CXX 480 CXXI 481 CXXII 487 CXXIII 488 CXXIV 489 CXXV 491 CXXVI 493 CXXVIII 495 CXXIX 498 CXXXI 499 CXXXII 501 CXXXIII 502 CXXXIV 503 CXXXV 504 CXXXVI 509 CXXXVII 513 CXXXVIII 514 CXXXIX 516 CXL 519 CXLI 529 CXLII 532 CXLIII 535 CXLIV 536 CXLV 537 CXLVI 540 CXLVII 545 CXLVIII 547 CXLIX 548 CL 549 CLI 552 CLII 554 CLIII 556 CLIV 561 CLV 562 CLVI 564 CLVII 566 CLVIII 568 CLIX 569 CLX 573 CLXI 574 CLXII 577 CLXIV 578 CLXV 580 CLXVI 582 CLXVII 584 CLXVIII 586 CLXIX 590 CLXX 592 CLXXII 593 CLXXIII 594 CLXXIV 599 CLXXV 600 CLXXVI 603 CLXXVII 610 CLXXVIII 612 Copyright