Borel set: Difference between revisions
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imported>Jitse Niesen m (copyedit: typography, remove empty sections, remove empty phrase "note that in the above") |
imported>Hendra I. Nurdin m (→Formal definition: link to topology-->link to topological space) |
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==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 σ-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 [[topological space|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 σ-algebra generated by <math>O</math>. | ||
The σ-algebra generated by <math>O</math> is simply the smallest σ-algebra containing the sets in <math>O</math> or, equivalently, the intersection of all σ-algebras containing <math>O</math>. | The σ-algebra generated by <math>O</math> is simply the smallest σ-algebra containing the sets in <math>O</math> or, equivalently, the intersection of all σ-algebras containing <math>O</math>. | ||
Revision as of 17:06, 16 September 2007
In mathematics, a Borel set is a set that belongs to the σ-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 σ-algebra generated by .
The σ-algebra generated by is simply the smallest σ-algebra containing the sets in or, equivalently, the intersection of all σ-algebras containing .