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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a -algebra (or sigma-algebra) over a set X is a nonempty collection of subsets of X (including X itself) that is closed under complementation and countable unions of its members. It is a Boolean algebra, completed to include countably infinite operations. A measure on X is a function which assigns a real number to subsets of X; this can be thought of as making precise a notion of "size" or "volume" for sets. One might like to assign such a size to every subset of X, but the axiom of choice implies that when the size under consideration is standard length for subsets of the real line, then there exist sets known as Vitali sets for which no size exists. For this reason, one considers instead a smaller collection of privileged subsets of X whose measure is defined; these sets constitute the -algebra.