This book has a long history. It began over two decades ago as the first half of a book then in progress on information and ergodic theory. The intent was and remains to provide a reasonably self-contained advanced (at least for engineers) treatment of measure theory, probability theory, and random processes, with an emphasis on general alphabets and on ergodic and stationary properties of random processes that might be neither ergodic nor stationary. The intended audience was mathematically inclined engineering graduate students and visiting scholars who had not had formal courses in measure theoretic probability or ergodic theory. Much of the material is familiar stuff for mathematicians, but many of the topics and results had not then previously appeared in books. Several topics still find little mention in the more applied literature, including descriptions of random processes as dynamical systems or flows; general alphabet models including standard spaces, Polish spaces, and Lebesgue spaces; nonstationary and nonergodic processes which have stationary and ergodic properties; metrics on random processes which quantify how different they are in ways useful for coding and signal processing; and stationary or sliding-block mappings - coding or filtering - of random processes.