Measure Integral and Probability serves as a gentle introduction to subject matter which is central to the foundations of analysis and probability. With an emphasis on clear explanations, and a step-by-step development of basic ideas, the level and style are ideal for advanced undergraduate students. The development of the Lebesgue integral provides the essential ideas; the role of basic convergence theorems, a discussion of modes of convergence for measurable functions, and relations to the Riemann Integral lead to the definition of Lebesgue spaces, the Fubini theorem and their roles in describing the properties of random variables and their distributions. With significant applications to probability, including laws of large numbers and the central limit theorem, this book will also be of interest to postgraduates needing to refresh or enhance their knowledge. Numerous exercises are provided, each with solutions.