- Borel algebra
mathematics, the Borel algebra (or Borel σ-algebra) on a topological space"X" is a σ-algebra of subsets of "X" associated with the topology of "X". In the mathematics literature, there are at least two "nonequivalent" definitions of this σ-algebra:
* The minimal σ-algebra containing the
* 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 of all σ-algebras containing "T" is itself a σ-algebra containing "T".
The elements of the Borel algebra are called Borel sets, and a subset of "X" which is a Borel set is called a Borel subset.
In general topological spaces, even
locally compactones, the two structures can be different, although this phenomenon is generally considered to be pathological in mathematical analysis. Indeed, the two structures are identical whenever the topological space is a locally compact separable metrizable space.
These algebras are named after
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 setP("X") of "X"), let
* be all countable unions of elements of "T"
* be all countable intersections of elements of "T"
transfinite inductiona 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 "G"ω1, where ω1 is the first uncountable ordinal number. 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.
An important example, especially in the theory of probability, is the Borel algebra on the set of
real numbers. It is the algebra on which the Borel measureis defined. Given a real random variable defined on a probability space, its probability distributionis 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 intervals.
In the construction by transfinite induction, it can be shown that, in each step, the number of sets is, at most, the
power of the continuum. So, the total number of Borel sets is less than or equal to .
tandard Borel spaces and Kuratowski theorems
The following is one of a number of theorems of
Kuratowskion Borel spaces:A Borel space is just another name for a set equipped with a distinguished σ-algebra; by extension elements of the distinguished σ-algebra are called Borel sets. Borel spaces form a category in which the maps are Borel measurable mappings between Borel spaces, where
is Borel measurable means that "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 isomorphicto one of(1) R, (2) Z or (3) a finite space.
Considered as Borel spaces, the real line R and the union of R with a countable set are isomorphic.
A standard Borel space is the Borel space associated to a Polish space.
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
probability measureon a standard Borel space turns it into a standard probability space.
Descriptive set theory
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
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