Borel set: Difference between revisions
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In [[mathematics]], a Borel set is a set that belongs to the <math>\sigma-</math>algebra (see the entry [[ | In [[mathematics]], a Borel set is a set that belongs to the <math>\sigma-</math>algebra (see the CZ entry [[sigma algebra]]) generated by the open sets of a [[topological space]]. | ||
==Formal definition== | ==Formal definition== | ||
Let <math>(X,O)</math> be a topological space, i.e. <math>X</math> is a set and <math>O</math> are the open sets of <math>X</math> (or, equivalently, the [[topology]] of <math>X</math>). Then <math>A \subset X </math> is a Borel set of <math>X</math> if <math>A \in \sigma(O) </math>, where <math>\sigma(O)</math> denotes the sigma algebra generated by <math>O</math>. | Let <math>(X,O)</math> be a topological space, i.e. <math>X</math> is a set and <math>O</math> are the open sets of <math>X</math> (or, equivalently, the [[topology]] of <math>X</math>). Then <math>A \subset X </math> is a Borel set of <math>X</math> if <math>A \in \sigma(O) </math>, where <math>\sigma(O)</math> denotes the sigma algebra generated by <math>O</math>. | ||
Note in the above that by definition <math>\sigma(O)</math> is simply the smallest <math>\sigma-</math>algebra containing the sets in <math>O</math> or, equivalently, the intersection of all <math>\sigma-</math>algebras containing <math>O</math>. | |||
== See also == | == See also == | ||
[[Topological space]] | |||
[[Sigma algebra]] | |||
[[Measure theory]] | |||
[[Probability theory]] | |||
Revision as of 05:34, 31 August 2007
In mathematics, a Borel set is a set that belongs to the algebra (see the CZ entry sigma algebra) generated by the open sets of a topological space.
Formal definition
Let be a topological space, i.e. is a set and are the open sets of (or, equivalently, the topology of ). Then is a Borel set of if , where denotes the sigma algebra generated by .
Note in the above that by definition is simply the smallest algebra containing the sets in or, equivalently, the intersection of all algebras containing .
See also