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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.