The Borel sigma-algebra of a topological space is the smallest sigma-algebra containing all its open sets. Its members are called Borel sets. It supplies a canonical measurable structure determined by the topology, without requiring a measure to be chosen. This construction connects topology with measure theory and provides the usual measurable sets on the real line and other familiar spaces. (web.math.princeton.edu)
Definition and construction
Let (X) be a set equipped with a topology (\tau). Its Borel sigma-algebra is denoted by [ \mathcal B(X)=\sigma(\tau) =\bigcap{\mathcal A:\mathcal A\text{ is a sigma-algebra on }X,\ \tau\subseteq\mathcal A}. ] “Smallest” means smallest under inclusion: every sigma-algebra containing (\tau) also contains (\mathcal B(X)). The intersection is well defined because the power set (\mathcal P(X)) is one such sigma-algebra. Intersections of sigma-algebras are again sigma-algebras. (web.math.princeton.edu)
Consequently, (\mathcal B(X)) contains (X) and the empty set, and is closed under complements, countable unions, and countable intersections. All closed sets are Borel because they are complements of open sets. Closed sets therefore generate the same sigma-algebra as open sets. These closure operations can be applied repeatedly, producing sets much more complicated than the initial generators. (ocw.mit.edu)
Countability is essential: a sigma-algebra need not be closed under arbitrary uncountable unions. Unlike a topology, it must allow complements and countable intersections, but need not allow every union. Thus a Borel sigma-algebra and the topology generating it are different kinds of structures. (personal.stevens.edu)
The real line and useful generators
On the real line with its usual topology, (\mathcal B(\mathbb R)) can be generated by all open intervals. It is sufficient to use intervals ((p,q)) with rational endpoints, since they form a countable basis and every open set is a countable union of them. Alternatively, [ \mathcal B(\mathbb R) =\sigma{(-\infty,a):a\in\mathbb R} =\sigma{(-\infty,a]:a\in\mathbb R}. ] These descriptions are useful because measurability can be checked on a generating family rather than on every Borel set individually. (users.math.msu.edu)
Every singleton in (\mathbb R) is closed, so every countable subset is Borel. In particular, (\mathbb Q) is Borel, and its complement, the irrational numbers, is Borel as well. This illustrates that a Borel set need be neither open nor closed. (personal.stevens.edu)
A countable union of closed sets is called an (F_\sigma) set; a countable intersection of open sets is called a (G_\delta) set. Both classes consist of Borel sets. For example, [ \mathbb Q=\bigcup_{q\in\mathbb Q}{q} ] is (F_\sigma), while (\mathbb R\setminus\mathbb Q) is (G_\delta). Further countable unions and intersections yield additional Borel sets. (web.math.princeton.edu)
Measurable functions and probability
A map (f:X\to Y) between topological spaces is Borel measurable when [ f^{-1}(A)\in\mathcal B(X) \quad\text{for every }A\in\mathcal B(Y). ] This is a particular case of a measurable function. Every continuous function is Borel measurable: inverse images of open sets are open, and inverse images preserve complements and countable unions. Borel measurability is weaker than continuity. For example, the indicator function of (\mathbb Q) is Borel measurable but discontinuous everywhere. (personal.stevens.edu)
For a real-valued function, it suffices to check that the sets ({x:f(x)<a}) are measurable for every real (a). The sigma-algebra on the domain need not itself be Borel. Accordingly, a real-valued random variable on a probability space ((\Omega,\mathcal F,P)) is a map measurable from ((\Omega,\mathcal F)) into ((\mathbb R,\mathcal B(\mathbb R))). Its distribution is the measure [ P_Z(A)=P(Z^{-1}(A)),\qquad A\in\mathcal B(\mathbb R). ] The Borel sigma-algebra specifies which sets of possible values have probabilities assigned by this distribution. (users.math.msu.edu)
Borel sets and Lebesgue measurability
Every Borel subset of (\mathbb R) is Lebesgue measurable, but the converse fails. The Lebesgue sigma-algebra is the completion of the Borel sigma-algebra with respect to Lebesgue measure: it additionally includes every subset of a Borel set of measure zero, together with the sets obtained through the required sigma-algebra operations. (ocw.mit.edu)
For example, the usual Cantor set is closed and has Lebesgue measure zero. Every subset of it is Lebesgue measurable, although some subsets are not Borel. Thus Lebesgue measure restricted to Borel sets is not complete. Completion depends on a particular measure, whereas the Borel sigma-algebra depends only on topology. (ocw.mit.edu)
Product spaces
For topological spaces (X) and (Y), the product sigma-algebra generated by Borel rectangles satisfies [ \mathcal B(X)\otimes\mathcal B(Y) \subseteq\mathcal B(X\times Y), ] where the Cartesian product has the product topology. Equality holds if both spaces have countable bases: every open subset of their product is then a countable union of open rectangles. Without suitable hypotheses, equality is not automatic. (math.vanderbilt.edu)
In particular, equality holds for finite or countable products of separable metric spaces. For Euclidean space, [ \mathcal B(\mathbb R^n)=\mathcal B(\mathbb R)^{\otimes n}. ] This identifies the measurable structure used for real-valued random vectors with the structure supplied by the ordinary Euclidean topology. (users.math.msu.edu)