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- The minimal σ-algebra containing the open sets.
- The minimal σ-algebra containing the compact sets.
The minimal σ-algebra on a set X containing a subset T of the power set 2X 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.
In general topological spaces, even locally compact ones, the two structures are different. They are however identical whenever the topological space is a locally compact separable metric space.
In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows: First define for any collection A of subsets of X (that is, for any subset of the power set P(X) of X),
Then define by transfinite recursion a sequence Gm, m an ordinal number, as follows:
- For the base case of the definition,
- If i is not a limit ordinal, then i has an immediately preceding ordinal i-1:
- If i is a limit ordinal,
Then the Borel algebra is Gm for the first uncountable ordinal number m.
To prove this fact, note that any open set in a metric space is the union of an incréasing sequence of closed sets. In particular, it is éasy to show that complementation of sets maps Gm into itself for any limit ordinal; moréover if m is an uncountable limit ordinal, Gm is closed under countable unions.
This alternate definition is useful for some set-théoretic considerations, but the minimalist definition is preferred by analysts.
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 réal random variable defined on a probability space, its probability distribution is by definition, also a méasure on the Borel algebra. The Borel algebra on the réals is the smallest sigma algebra on R which contains all the intervals.
The following is one of a number of Kuratowski théorems 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 méasurable mappings between Borel spaces, where f:X -> Y is Borel méasurable iff f-1(B) is Borel in X for any Borel subset B of Y.
Théorem. 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.
An excellent exposition of the machinery of Polish topology is given in Chapter 3 of the following reference: