Editing Borel set (section)

== Generating the Borel algebra ==

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
* <math>T_\sigma </math> be all countable unions of elements of ''T''
* <math>T_\delta </math> be all countable intersections of elements of ''T''
* <math>T_{\delta\sigma} = (T_\delta)_\sigma.</math>

Now define by [[transfinite induction]] a sequence ''G<sup>m</sup>'', where ''m'' is an [[ordinal number]], in the following manner:
* For the base case of the definition, let <math> G^0</math> be the collection of open subsets of ''X''.
* If ''i'' is not a [[limit ordinal]], then ''i'' has an immediately preceding ordinal ''i'' − 1. Let <math display="block"> G^i = [G^{i-1}]_{\delta \sigma}.</math>
* If ''i'' is a limit ordinal, set <math display="block"> G^i = \bigcup_{j < i} G^j. </math>

The claim is that the Borel algebra is ''G''<sup>ω<sub>1</sub></sup>, where ω<sub>1</sub> is the [[first uncountable ordinal|first uncountable ordinal number]]. That is, the Borel algebra can be ''generated'' from the class of open sets by iterating the operation
<math display="block"> G \mapsto G_{\delta \sigma}. </math>
to the first uncountable ordinal.

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 ''G<sup>m</sup>'' into itself for any limit ordinal  ''m''; moreover if ''m'' is an uncountable limit ordinal, ''G<sup>m</sup>'' is closed under countable unions.

For each Borel set ''B'', there is some countable ordinal ''α<sub>B</sub>'' such that ''B'' can be obtained by iterating the operation over ''α<sub>B</sub>''. However, as ''B'' varies over all Borel sets, ''α<sub>B</sub>'' will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ''ω''<sub>1</sub>, the first uncountable ordinal.

The resulting sequence of sets is termed the [[Borel hierarchy]].

=== Example ===
An important example, especially in the [[probability theory|theory of probability]], is the Borel algebra on the set of [[real number]]s. It is the algebra on which the [[Borel measure]] is defined.  Given a [[Random variable#Real-valued random variables|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 [[interval (mathematics)|intervals]].

In the construction by transfinite induction, it can be shown that, in each step, the [[cardinality|number]] of sets is, at most, the [[cardinality of the continuum]]. So, the total number of Borel sets is less than or equal to
<math display="block">\aleph_1 \cdot 2 ^ {\aleph_0}\, = 2^{\aleph_0}.</math>

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 <math>2^{2^{\aleph_0}}</math>).