Suitable for a graduate course in analytic probability theory, this text requires no previous knowledge of probability and only a limited background in real analysis. In addition to providing instruction for graduate students in mathematics and mathematical statistics, the book features detailed proofs that offer direct access to the basic theorems of probability theory for mathematicians of all interests. The treatment strikes a balance between measure-theoretic aspects of probability and distribution aspects, presenting some of the basic theorems of analytic probability theory in a cohesive manner. Statements are rendered as simply as possible in order to make them easy to remember and to demonstrate the essential idea behind each proof. Topics include probability spaces and distributions, stochastic independence, basic limiting operations, strong limit theorems for independent random variables, the central limit theorem, conditional expectation and Martingale theory, and an introduction to stochastic processes, particularly Brownian motion. Each section concludes with problems that reinforce the preceding material.