This introduction can be used, at the beginning graduate level, for a one-semester course on probability theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as finance theory, electrical engineering, and operations research. The text covers the essentials in a directed and lean way with 28 short chapters, and assumes only an undergraduate background in mathematics. Readers are taken right up to a knowledge of the basics of Martingale Theory, and the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference.

We have made small changes throughout the book, including the exercises, and we have tried to correct if not all, then at least most of the typos. We wish to thank the many colleagues and students who have commented c- structively on the book since its publication two years ago, and in particular Professors Valentin Petrov, Esko Valkeila, Volker Priebe, and Frank Knight. Jean Jacod, Paris Philip Protter, Ithaca March, 2002 Preface to the Second Printing of the Second Edition We have bene?ted greatly from the long list of typos and small suggestions sent to us by Professor Luis Tenorio. These corrections have improved the book in subtle yet important ways, and the authors are most grateful to him. Jean Jacod, Paris Philip Protter, Ithaca January, 2004 Preface to the First Edition We present here a one semester course on Probability Theory. We also treat measure theory and Lebesgue integration, concentrating on those aspects which are especially germane to the study of Probability Theory. The book is intended to ?ll a current need: there are mathematically sophisticated s- dents and researchers (especially in Engineering, Economics, and Statistics) who need a proper grounding in Probability in order to pursue their primary interests. Many Probability texts available today are celebrations of Pr- ability Theory, containing treatments of fascinating topics to be sure, but nevertheless they make it di?cult to construct a lean one semester course that covers (what we believe) are the essential topics.

1. Introduction .............................................. 1 2. Axioms of Probability .................................... 7 3. Conditional Probability and Independence ................ 15 4. Probabilities on a Finite or Countable Space.............. 21 5. Random Variables on a Countable Space ................. 27 6. Construction of a Probability Measure.................... 35 7. Construction of a Probability Measure on R .............. 39 8. Random Variables ........................................ 47 9. Integration with Respect to a Probability Measure ....... 51 10. Independent Random Variables ........................... 65 11. Probability Distributions on R............................ 77 12. Probability Distributions on Rn .......................... 87 13. Characteristic Functions .................................. 103 14. Properties of Characteristic Functions .................... 111 15. Sums of Independent Random Variables .................. 117 16. Gaussian Random Variables (The Normaland the Multivariate Normal Distributions) .............. 125 17. Convergence of Random Variables ........................ 141 18. Weak Convergence ....................................... 151 19. Weak Convergence and Characteristic Functions .......... 167 20. The Laws of Large Numbers .............................. 173 21. The Central Limit Theorem .............................. 181 22. L2 and Hilbert Spaces .................................... 189 23. Conditional Expectation .................................. 197 24. Martingales ............................................... 211 25. Supermartingales and Submartingales .................... 219 26. Martingale Inequalities ................................... 223 27. Martingale Convergence Theorems ....................... 229 28. The Radon-Nikodym Theorem............................ 243 References .................................................... 249 Index ......................................................... 251