Lemma: Let C = {(a; b): a < b}. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets. The smallest algebra containing C, a collection of subsets of a set X, is called the algebra generated by C. Definition. 1.2 Generated Sigma-algebra ˙(B) Let X be a set and B a non-empty collection of subsets of X. A. Note that this σ-algebra is not, in general, the whole power set. Remark 0.1 It follows from the de nition that a countable intersection of sets in Ais also in A. A sigma-algebra which is related to the topology of a set. $\endgroup$ – Stefan Geschke Sep 24 '10 at 18:39

Its elements are called Borel sets. That is, if O denotes the collection of all open subsets of R,thenB = σ(O). ˙{Algebras.

Let X = R and A = {A ⊂ R | A is finite or A˜ is finite}. De nition 0.1 A collection Aof subsets of a set Xis a ˙-algebra provided that (1) ;2A, (2) if A2Athen its complement is in A, and (3) a countable union of sets in Ais also in A. $\begingroup$ Your example is the sum of two copies of the countable/co-countable $\sigma$-algebra, which is the Borel algebra of the co-countable topology. Then A is an algebra but not a σ-algebra (since N = ∪{n} but N ∈ A/).

One can build up the Borel sets from the open sets by iterating the operations of complementation and taking countable unions.


Example. In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions. The smallest ˙{algebra containing all the sets of B is denoted ˙(B) and is called the sigma-algebra generated by the collection B. Definition: Borel σ-algebra (Emile Borel (1871-1956), France.) Given a topological space $X$, the Borel σ-algebraof $X$ is the $\sigma$-algebra generated by the open sets (i.e. A which contains C. That is, if B is any algebra containing C, then B contains A. Definition. The Borel sigma-algebra is defined to be the sigma-algebra generated by the open sets (or equivalently, by the closed sets). In fact, there is no simple procedure An algebra A of sets is a σ-algebra (or a Borel field) if every union of a countable collection of sets in A is again in A. Sigma Algebras and Borel Sets. An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

It turns out that the set H constructed in Lecture #4 is non-Borel, although we will not prove this at present. Lecture 5: Borel Sets Topologically, the Borel sets in a topological space are the σ-algebra generated by the open sets. the smallest $\sigma$-algebra of When $X$ is a locally compact Hausdorff space some authors define the Borel sets as the smallest $\sigma$-ringcontaining the compact sets, see [Hal]. De nition 0.2 Let fA ng1 Proposition 1.13. The Borel σ-algebra (or, Borel field) denoted B, of the topological space (X; τ) is the σ-algebra generated by the family τof open sets. That is, if O denotes the collection of all open subsets of R,thenB = σ(O). In this video, I introduce sigma algebras, generating sigma algebras, the Borel sigma algebra, and much more. Borel sets are named after Émile Borel. ... proved that there exist non-Borel sets.