This book presents a graduate course in measure-theoretic probability designed to cover a one year course at a comfortable pace. The author starts with classical probability (including a neat proof of the de Moivre-Laplace theorem) before moving on to elements of measure theory (measure spaces, integration, modes of convergence and product spaces). The chapter on independence covers the strong law of large numbers and the Glivenko-Cantalli theorem. The central limit theorem in an independent and identically distributed setting is proved both by the classical harmonic-analytic approach and by the Liapunov-Lindeberg replacement method. The material also covers elements of weak convergence and the Cramer characterization of normal distributions.
Some rich material on conditional expectations and discrete time martingales is offered, the optional stopping theorem and the martingale convergence theorem being the principal results. The applications range from the Lebesgue differentiation theorem to option-pricing procedures. Brownian motion is constructed, its nowhere differentiability and the strong Markov property being the principal achievements. The text terminates with a short introduction to stochastic calculus (the Itô integral and formula, optional stopping, L2-martingales and the exit distribution of Brownian motion). The reader might miss a treatment of Markov chains but overall the text is neatly written and presents not only good mathematics but also a heuristic view of modern probability, including numerous exercises and historical notes.