Measure and Probability
This book is designed as an introduction to probabilistic measure theory for MSc and PhD students oriented to stochastics. A standard treatment of the fundamentals of measure theory (measure and its extension, measurability and integration) is presented in chapter 1 and chapter 2. Useful probabilistic specifications, e.g. random variables, their distribution and (conditional) expectation, and independent events and variables, are discussed in chapter 3. Chapter 4 goes as far as Kolmogorov's zero-one law and consistency theorem. Chapter 5 treats characteristic functions calculus, modes of convergence (for R-valued variables) and ends with the central limit theorem and the law of large numbers, the proof of which is based on the ergodic theorem for a stationary sequence of random variables. Chapter 6 focuses on discrete-time Markov chains with a countable state space. The basic notions (aperiodicity, irreducibility, transience, recurrence and stationarity) are treated with a balanced mixture of intuition and mathematical rigour. Chapter 7 completes the text with Lp-spaces, the Radon-Nikodym theorem, differentiation and change of variables rules and, finally, the Riesz representation theorem. The book is neatly written and can be recommended as an introduction to all students who intend to start courses on advanced modern probability.