Rings of sets

Abstract

It is a familiar fact of elementary calculus that the integral of a function exists only if the function is continuous, or nearly so. In the theory of the Lebesgue integral, with which we are concerned in this book, continuity is replaced by a significantly less stringent requirement known as measurability. This concept, in turn, is defined in terms of a certain type of collection of sets, called a $\sigma$ -algebra, and so we begin with a brief look at this and some related concepts.