Measure Theory for Analysis and Probability

This book covers major measure theory topics with a fairly extensive study of their applications to probability and analysis. It begins by demonstrating the essential nature of measure theory before delving into the construction of measures and the development of integration theory. Special attention is given to probability spaces and random variables/vectors. The text then explores product spaces, Radon–Nikodym and Jordan–Hahn theorems, providing a detailed account of   spaces and their duals. After revisiting probability theory, it discusses standard limit theorems such as the laws of large numbers and the central limit theorem, with detailed treatment of weak convergence and the role of characteristic functions.

The book further explores conditional probabilities and expectations, preceded by motivating discussions. It discusses the construction of probability measures on infinite product spaces, presenting Tulcea’s theorem and Kolmogorov’s consistency theorem. The text concludes with the construction of Brownian motion, examining its path properties and the significant strong Markov property. This comprehensive guide is invaluable not only for those pursuing probability theory seriously but also for those seeking a robust foundation in measure theory to advance in modern analysis. By effectively motivating readers, it underscores the critical role of measure theory in grasping fundamental probability concepts.

1. Measure Theory: Why and What.- 2. Measures: Construction and Properties.- 3. Measurable Functions and Integration.- 4. Random Variables and Random Vectors.- 5. Product Spaces.- 6. Radon-Nikodym Theorem and Lp Spaces.- 7. Convergence and Laws of Large Numbers.- 8. Weak convergence and Central Limit Theorem.- 9. Conditioning: The Right Approach.- 10. Infinite Products.- 11. Brownian Motion: A Brief Journey.

Alok Goswami is Visiting Professor in the School of Mathematical and Computational Science at the Indian Association for the Cultivation of Science, Kolkata, West Bengal India. Earlier, he was a faculty member in the Theoretical Statistics and Mathematics Unit at the Indian Statistical Institute, West Bengal, Kolkata, India, for nearly 30 years, until his retirement in 2019, as Professor. He received his M.Stat. degree from the Indian Statistical Institute in 1980 and his Ph.D. degree from the Department of Mathematics in the University of Illinois at Urbana-Champaign, U.S.A., in 1986. He visited many other premier universities and institutes of India and abroad and is widely regarded as one of the top academicians in the field of probability theory. His primary research interests are probability and stochastic processes, while he has also done some collaborative research in statistics. 

B.V. Rao has been at Chennai Mathematical Institute, Tamil Nadu, India, since 2009, and is currently a Visiting Faculty there. Earlier, he served as a faculty member at the Indian Statistical Institute, Kolkata, for more than 30 years, before retiring as a Distinguished Professor in 2009. He received his M.Sc. in Statistics from Osmania University, Hyderabad, India, in 1965, and his Ph.D. from the Indian Statistical Institute, Kolkata, in 1970. His research interests include descriptive set theory, stochastic processes, characterization of distributions and statistics, percolation theory, iterated function systems and spin glasses. He has lectured at several places in India and abroad.