Measure Theory, Probability, and Stochastic Processes

Jean-François ​Le Gall is Professor of Mathematics at the University of Paris-Saclay in France. As one of the leading experts in probability theory, he has done extensive research on stochastic processes, including Brownian motion, random trees, random planar maps, and other related objects. His research accomplishments have been recognized with various awards, most recently the Wolf prize. He is the author of two successful textbooks on Brownian Motion, Martingales, and Stochastic Calculus (2016) in the Graduate Texts in Mathematics series and Spatial Branching Processes, Random Snakes and Partial Differential Equations (1999) in the Lectures in Mathematics, ETH Zürich series.

“This book is a good choice for a course introducing both measure theory and probability. It would also be particularly good for self-study by those wishing to see how ideas from measure theory affect our understanding of probability and vice versa. There are exercises at the end of each chapter: some are relatively straightforward, while others expand on topics mentioned briefly in the text.” (Thomas Polaski, Mathematical Reviews, August, 2024)“This book covers probability theory and stochastic processes at a graduate level. … this book is exceptionally well written in a concise manner and is suitable for individuals with a strong background in undergraduate real analysis and undergraduate probability.” (Eunghyun Lee, zbMATH 1526.60001, 2024)

• Part I. Measure Theory.- Chapter 1. Measurable Spaces.- Chapter 2. Integration of Measurable Functions.- Chapter 3. Construction of Measures.- Chapter 4. Lp Spaces.- Chapter 5. Product Measure.- Chapter 6. Signed Measures.- Chapter 7. Change of Variables.- Part II. Probability Theory.- Chapter 8. Foundations of Probability Theory.- Chapter 9. Independence.- Chapter 10. Convergence of Random Variables.- Chapter 11. Conditioning.- Part III. Stochastic Processes.- Chapter 12. Theory of Martingales.- Chapter 13. Markov Chains.- Chapter 14. Brownian Motion.