Aljabar Borel

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Dina matematik, aljabar Borel (atawa Borel σ-aljabar) dina ruang topologi nyaeta dua σ-aljabar sejen dina topologi ruang X:

  • The minimal σ-algebra containing the open sets.
  • The minimal σ-algebra containing the compact sets.




 A_\sigma = \mbox{ countable unions of elements of } A \quad
 A_\delta = \mbox{ countable intersections of elements of } A \quad


  • For the base case of the definition,
 G^0 = \mbox{ open subsets of } X
  • If i is a limit ordinal,


To prove this fact, note that any open set in a metric space is the union of an increasing sequence of closed sets. In particular, it is easy to show that complementation of sets maps Gm into itself for any limit ordinal; moreover if m is an uncountable limit ordinal, Gm is closed under countable unions.

This alternate definition is useful for some set-theoretic considerations, but the minimalist definition is preferred by analysts.

Examples[édit | édit sumber]

A particularly important example is the Borel sigma algebra (or just Borel algebra) on the set of real numbers. It is the algebra on which the Borel measure is defined. Given a real random variable defined on a probability space, its probability distribution is by definition, also a measure on the Borel algebra. The Borel algebra on the reals is the smallest sigma algebra on R which contains all the intervals.

The following is one of a number of Kuratowski theorems on Borel spaces: A Borel space is just another name for a set equipped with a σ-algebra. Borel spaces form a category in which the maps are Borel measurable mappings between Borel spaces, where f:X -> Y is Borel measurable iff f-1(B) is Borel in X for any Borel subset B of Y.




See also[édit | édit sumber]

Baire set

Rujukan[édit | édit sumber]


  • Paul Halmos, Measure Theory, D.van Nostrand Co., 1950