In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space ''X'' is either of the two sigma-algebras: * The minimal σ-algebra containing the open sets. * The minimal σ-algebra containing the compact sets. Here, the minimal σ-algebra containing a collection ''T'' of subsets of ''X'' is the smallest σ-algebra containing ''T''. The existence and uniqueness of the minimal σ-algebra is shown by noting that the intersection (set theory) of all σ-algebras containing ''T'' is itself a σ-algebra containing ''T''. The elements of the Borel algebra are called Borel sets. In general topological spaces, even locally compact ones, the two structures can be different. They are however identical whenever the topological space is a locally compact separable space metric space. == Generating the Borel algebra == In the case ''X'' is a metric space, the Borel algebra in the first sense may be described ''generatively'' as follows. For a collection ''T'' of subsets of ''X'' (that is, for any subset of the power set P(''X'') of ''X''), let * be all countable unions of elements of ''T'' * be all countable intersections of elements of ''T'' * Define by transfinite induction a sequence ''Gm'', where ''m'' is an ordinal number, in the following manner: * For the base case of the definition, : = the collection of open subsets of ''X''. * If ''i'' is not a limit ordinal, then ''i'' has an immediately preceding ordinal ''i − 1''. Let : * If ''i'' is a limit ordinal, set : We now claim that the Borel algebra is ''Gm'' for the first uncountable ordinal number ''m'', that is, the Borel algebra can be ''generated'' from the class of open sets by iterating the operation : to the first uncountable ordinal. (Note: for any fixed Borel set, we only have to iterate a countable number of times, but as we vary across all Borel sets, this countable number of times is arbitrarily large and approaches the first uncountable 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 == An important example, especially in the probability theory, is the 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 σ-algebra on R which contains all the interval (mathematics). 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''. Theorem. Let ''X'' be a Polish space, that is a topological space such that there is a metric ''d'' on ''X'' which defines the topology of ''X'' and which makes ''X'' a complete separable metric space. Then ''X'' as a Borel space is isomorphic to one of (1) R, (2) Z or (3) a finite space. It should be noted that as Borel spaces R and R union with a countable set, are isomorphic. For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel. See analytic set. ==See also== * Baire set == References == An excellent exposition of the machinery of ''Polish topology'' is given in Chapter 3 of the following reference: * William Arveson, ''An Invitation to C*-algebras'', Springer-Verlag, 1981 * Richard Dudley, '' Real Analysis and Probability''. Wadsworth, Brooks and Cole, 1989 * Paul Halmos, ''Measure Theory'', D.van Nostrand Co., 1950 * Halsey Royden, ''Real Analysis'', Prentice Hall, 1988 Topology
''A subset of X is a Borel set if and only if it can be obtained from open sets by using a countable series of the set operations union, intersection and complement.'' Is this true in general? I know that for metrisable spaces, every Borel set is of the form F-sigma, G-delta, F-sigma-delta, G-delta-sigma, F-sigma-delta-sigma, etc., etc. But I don't think every Borel set is necessarily of this form in a general topological space. (Why else would metrisability be part of the hypothesis of the theorem I read?) :I was quite wrong here. I read the symbol for the first uncountable ordinal as the symbol for the first infinite ordinal. Whoops! User:Revolver 07:03, 16 Dec 2004 (UTC) In any case, I'm not sure it's clear precisely what is meant by "using a countable series of the set operations union, intersection, and complement". Certainly this is meant to include the F-sigma, G-delta, F-sigma-delta, G-delta-sigma, F-sigma-delta-sigma, etc., etc., but what about taking countable unions/intersections of sets lying somewhere in this hierarchy? I'm not even sure if this gives more sets, but regardless, the way it's worded, it's not clear if this construction is intended or not. user:revolver :: I believe The article is true as it now stands with your correction. I may have been responsible for the original sloppy wording.User:CSTAR 13:14, 16 Dec 2004 (UTC) I noticed that this article is virtually identical to [http://www.tutorgig.com/encyclopedia/getdefn.jsp?keywords=Borel_algebra]. Did they copy from us, or the other way around? If it's the latter, it would seem to be a violation. user:revolver Sorry...it was copied from us! We're given credit at the bottom of the page. My bad. user:revolver == generation of Borel sets == I didn't mean iteration to any countable ordinal. I meant that any Borel set could be created by iteration to a countable ordinal, and that this countable ordinal may be arbitrarily large depending on the Borel set. User:Revolver 19:31, 14 Jun 2005 (UTC) :I just didn't want to give the impression that there is a Borel set whose existence requires one to iterate ''uncountably many times''. To get the whole ''algebra'', you must go to uncountably many times, but not for a fixed Borel set. User:Revolver 19:40, 14 Jun 2005 (UTC) :: Ah, yes.--User:CSTAR 20:17, 14 Jun 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Borel_algebra:
Borel_algebra
Borel_algebra
Borel algebra
In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space ''X'' is either of the two sigma-algebras: * The minimal σ-algebra containing the open sets. * The minimal σ-algebra containing the compact sets. Here, the minimal σ-algebra containing a collection ''T'' of subsets of ''X'' is the smallest σ-algebra containing ''T''. The existence and uniqueness of the minimal σ-algebra is shown by noting that the intersection (set theory) of all σ-algebras containing ''T'' is itself a σ-algebra containing ''T''. The elements of the Borel algebra are called Borel sets. In general topological spaces, even locally compact ones, the two structures can be different. They are however identical whenever the topological space is a locally compact separable space metric space. == Generating the Borel algebra == In the case ''X'' is a metric space, the Borel algebra in the first sense may be described ''generatively'' as follows. For a collection ''T'' of subsets of ''X'' (that is, for any subset of the power set P(''X'') of ''X''), let * be all countable unions of elements of ''T'' * be all countable intersections of elements of ''T'' * Define by transfinite induction a sequence ''Gm'', where ''m'' is an ordinal number, in the following manner: * For the base case of the definition, : = the collection of open subsets of ''X''. * If ''i'' is not a limit ordinal, then ''i'' has an immediately preceding ordinal ''i − 1''. Let : * If ''i'' is a limit ordinal, set : We now claim that the Borel algebra is ''Gm'' for the first uncountable ordinal number ''m'', that is, the Borel algebra can be ''generated'' from the class of open sets by iterating the operation : to the first uncountable ordinal. (Note: for any fixed Borel set, we only have to iterate a countable number of times, but as we vary across all Borel sets, this countable number of times is arbitrarily large and approaches the first uncountable 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 == An important example, especially in the probability theory, is the 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 σ-algebra on R which contains all the interval (mathematics). 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''. Theorem. Let ''X'' be a Polish space, that is a topological space such that there is a metric ''d'' on ''X'' which defines the topology of ''X'' and which makes ''X'' a complete separable metric space. Then ''X'' as a Borel space is isomorphic to one of (1) R, (2) Z or (3) a finite space. It should be noted that as Borel spaces R and R union with a countable set, are isomorphic. For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel. See analytic set. ==See also== * Baire set == References == An excellent exposition of the machinery of ''Polish topology'' is given in Chapter 3 of the following reference: * William Arveson, ''An Invitation to C*-algebras'', Springer-Verlag, 1981 * Richard Dudley, '' Real Analysis and Probability''. Wadsworth, Brooks and Cole, 1989 * Paul Halmos, ''Measure Theory'', D.van Nostrand Co., 1950 * Halsey Royden, ''Real Analysis'', Prentice Hall, 1988 Topology
Borel algebra
''A subset of X is a Borel set if and only if it can be obtained from open sets by using a countable series of the set operations union, intersection and complement.'' Is this true in general? I know that for metrisable spaces, every Borel set is of the form F-sigma, G-delta, F-sigma-delta, G-delta-sigma, F-sigma-delta-sigma, etc., etc. But I don't think every Borel set is necessarily of this form in a general topological space. (Why else would metrisability be part of the hypothesis of the theorem I read?) :I was quite wrong here. I read the symbol for the first uncountable ordinal as the symbol for the first infinite ordinal. Whoops! User:Revolver 07:03, 16 Dec 2004 (UTC) In any case, I'm not sure it's clear precisely what is meant by "using a countable series of the set operations union, intersection, and complement". Certainly this is meant to include the F-sigma, G-delta, F-sigma-delta, G-delta-sigma, F-sigma-delta-sigma, etc., etc., but what about taking countable unions/intersections of sets lying somewhere in this hierarchy? I'm not even sure if this gives more sets, but regardless, the way it's worded, it's not clear if this construction is intended or not. user:revolver :: I believe The article is true as it now stands with your correction. I may have been responsible for the original sloppy wording.User:CSTAR 13:14, 16 Dec 2004 (UTC) I noticed that this article is virtually identical to [http://www.tutorgig.com/encyclopedia/getdefn.jsp?keywords=Borel_algebra]. Did they copy from us, or the other way around? If it's the latter, it would seem to be a violation. user:revolver Sorry...it was copied from us! We're given credit at the bottom of the page. My bad. user:revolver == generation of Borel sets == I didn't mean iteration to any countable ordinal. I meant that any Borel set could be created by iteration to a countable ordinal, and that this countable ordinal may be arbitrarily large depending on the Borel set. User:Revolver 19:31, 14 Jun 2005 (UTC) :I just didn't want to give the impression that there is a Borel set whose existence requires one to iterate ''uncountably many times''. To get the whole ''algebra'', you must go to uncountably many times, but not for a fixed Borel set. User:Revolver 19:40, 14 Jun 2005 (UTC) :: Ah, yes.--User:CSTAR 20:17, 14 Jun 2005 (UTC)
See other meanings of words starting from letter:
B
BA | BC | BD | BE | BF | BG | BH | BI | BJ | BK | BL | BM | BN | BO | BP | BR | BS | BT | BU | BW | BX | BY | BZ |Words begining with Borel_algebra:
Borel_algebra
Borel_algebra
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