Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.
In the case that 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 Now define by transfinite induction a sequence Gm, where m is an ordinal number, in the following manner: The claim is that the Borel algebra is Gω1, 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 prove this claim, any open set in a metric space is the union of an increasing sequence of closed sets.
In particular, complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountable limit ordinal, Gm is closed under countable unions.
The resulting sequence of sets is termed the Borel hierarchy.
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 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 that 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 cardinality of the continuum.
So, the total number of Borel sets is less than or equal to
In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of Lebesgue measurable sets that exist, which is strictly larger and equal to
"[1] However, modern usage is to call the distinguished sub-algebra the measurable sets and such spaces measurable spaces.
The reason for this distinction is that the Borel sets are the σ-algebra generated by open sets (of a topological space), whereas Mackey's definition refers to a set equipped with an arbitrary σ-algebra.
Then X as a Borel space is isomorphic to one of (This result is reminiscent of Maharam's theorem.)
Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic.
A standard Borel space is characterized up to isomorphism by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.
For subsets of Polish spaces, Borel sets can be characterized as those sets that 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.
Every probability measure on a standard Borel space turns it into a standard probability space.
An example of a subset of the reals that is non-Borel, due to Lusin,[4] is described below.
Every irrational number has a unique representation by an infinite simple continued fraction where
be the set of all irrational numbers that correspond to sequences
It's important to note, that while Zermelo–Fraenkel axioms (ZF) are sufficient to formalize the construction of
However, this is a proof of existence (via the axiom of choice), not an explicit example.
According to Paul Halmos,[6] a subset of a locally compact Hausdorff topological space is called a Borel set if it belongs to the smallest σ-ring containing all compact sets.
Norberg and Vervaat[7] redefine the Borel algebra of a topological space
is second countable or if every compact saturated subset is closed (which is the case in particular if