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In mathematics, a **Borel set** is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

For a topological space *X*, the collection of all Borel sets on *X* forms a σ-algebra, known as the **Borel algebra** or **Borel σ-algebra**. The Borel algebra on *X* is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Borel_set

In mathematical analysis and in probability theory, a **σ-algebra** (also **sigma-algebra**, **σ-field**, **sigma-field**) on a set *X* is a collection Σ of subsets of *X* that is closed under countable-fold set operations (complement, union of countably many sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under *finitely many* set operations. That is, a σ-algebra 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**.

The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. 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. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Sigma-algebra

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