Editing Borel set (section)

=== 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>).