This text provides undergraduate mathematics students with an introduction to the modern theory of probability as well as the roots of the theory's mathematical ideas and techniques. Centered around the concept of measure and integration, the treatment is applicable to other branches of analysis and explores more specialized topics, including convergence theorems and random sequences and functions.The initial part is devoted to an exploration of measure and integration from first principles, including sets and set functions, general theory, and integrals of functions of real variables. These topics provide tools for use in the second part, which emphasizes underlying mathematical ideas, including the roles of random variables and limit processes in probability. The concise format and exposition offer an ideal review of the subject for students with some background in probability.