Borel set

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Generating the Borel algebra[edit]



  • T_\sigma \quad be all countable unions of elements of T
  • T_\delta \quad be all countable intersections of elements of T
  •  T_{\delta\sigma}=(T_\delta)_\sigma.\,


  • For the base case of the definition, let  G^0 be the collection of open subsets of X.
     G^i = [G^{i-1}]_{\delta \sigma}.
  • If i is a limit ordinal, set
     G^i = \bigcup_{j < i} G^j.


 G \mapsto G_{\delta \sigma}.

to the first uncountable ordinal.






In the construction by transfinite induction, it can be shown that, in each step, the number of sets is, at most, the power of the continuum. So, the total number of Borel sets is less than or equal to

\aleph_1 \times 2 ^ {\aleph_0}\, = 2^{\aleph_0}.\,

Standard Borel spaces and Kuratowski theorems[edit]

Mackey writes that a Borel space is "a set together with a distinguished σ-field of subsets called its Borel sets." [1] However, more modern terminology is to call such spaces measurable spaces. The reason for this distinction is that the Borel σ-algebra is 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. There exist measurable spaces which are not Borel spaces in this more restricted topological sense.[2]

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. A function f:X \rightarrow Y is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, f^{-1}(B) is a measurable set in X.

Theorem. 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. (This result is reminiscent of Maharam's theorem.)

Considered as Borel spaces, the real line R and the union of R with a countable set are isomorphic.


Any standard Borel space is defined (up to isomorphism) by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.



Non-Borel sets[edit]

An example of a subset of the reals which is non-Borel, due to Lusin[4] (see Sect. 62, pages 76–78), is described below. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.

Every irrational number has a unique representation by a continued fraction


where a_0\, is some integer and all the other numbers a_k\, are positive integers. Let A\, be the set of all irrational numbers that correspond to sequences (a_0,a_1,\dots)\, with the following property: there exists an infinite subsequence (a_{k_0},a_{k_1},\dots)\, such that each element is a divisor of the next element. This set A\, is not Borel. In fact, it is analytic, and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris, especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.

Another non-Borel set is an inverse image f^{-1}[0] of an infinite parity function f\colon \{0, 1\}^{\omega} \to \{0, 1\}. However, this is a proof of existence (via the axiom of choice), not an explicit example.

Alternative non-equivalent definitions[edit]

According to Halmos (Halmos 1950, page 219), 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 [5] redefine the Borel algebra of a topological space X as the \sigma–algebra generated by its open subsets and its compact saturated subsets. This definition is well-suited for applications in the case where X is not Hausdorff. It coincides with the usual definition if X is second countable or if every compact saturated subset is closed (which is the case in particular if X is Hausdorff).

See also[edit]


An excellent exposition of the machinery of Polish topology is given in Chapter 3 of the following reference:



  2. ^ Jochen Wengenroth (, Is every sigma-algebra the Borel algebra of a topology?, (version: 2012-02-09)
  3. ^ Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7 
  4. ^ Lusin, Nicolas (1927), "Sur les ensembles analytiques", Fundamenta Mathematicae (Institute of mathematics, Polish academy of sciences) 10: 1–95 .

External links[edit]