The authors believe that a proper treatment of probability theory requires an adequate background in the theory of finite measures in general spaces. The first part of their book sets out this material in a form that not only provides an introduction for intending specialists in measure theory but also meets the needs of students of probability. The theory of measure and integration is presented for general spaces, with Lebesgue measure and the Lebesgue integral considered as important examples whose special properties are obtained. The introduction to functional analysis which follows covers the material (such as the various notions of convergence) which is relevant to probability theory and also the basic theory of L2-spaces, important in modern physics. The second part of the book is an account of the fundamental theoretical ideas which underlie the applications of probability in statistics and elsewhere, developed from the results obtained in the first part. A large number of examples is included; these form an essential part of the development.