Sigma-algebra

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In mathematics, a σ-algebra (also sigma-algebra, σ-field, sigma-field) is a technical concept for a collection of sets satisfying certain properties.[1] The main use of σ-algebras is in the definition of measures; specifically, a σ-algebra is the collection of sets over which a measure is defined. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities.

By definition, a σ-algebra over a set X is a nonempty collection Σ of subsets of X (including X itself) that is closed under complementation and countable unions of its members. It is an algebra of sets, completed to include countably infinite operations. The pair (X, Σ) is also a field of sets, called a measurable space.

Thus, if X = {a, b, c, d}, one possible sigma algebra on X is

Contents

Motivation

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 would like to assign such a size to every subset of X, but in many natural settings, this is not possible. For example the axiom of choice implies that when the size under consideration is the standard notion of 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, which will be called the measurable sets, and which are closed under operations that one would expect for measurable sets, that is the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.

Definition and properties

Let X be some set, and 2X its power set. Then a subset Σ ⊂ 2X is called σ-algebra if it satisfies the following three properties:

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