In 1933 the Russian mathematician A. N. Kolmogorov wrote the famous book Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of Probability) in which he formulated the basic axioms of probability theory. The appearance of Measure Theory by P. R. Halmos and An Introduction to Probability Theory and Its Applications by W. Feller in 1950 made both subjects accessible to all undergraduate and graduate students in mathematics all over the world. The present book has been written in the hope that it will provide the impetus to introduce in undergraduate and graduate programmes both measure theory and probability theory as a one-year course. Since the study of probability theory in its advanced stage depends on a knowledge of measure theory, special effort has been made to integrate the two subjects into a single volume. The material of the book grew out of the lectures delivered by the author to M. Sc. students at the Centre of Advanced Study in math- ematics in the University of Bombay, M. Stat. students of the Indian Statistical Institute, Delhi, and M. Sc. students at the Indian Institute of Technology, Delhi.