Borel hierarchy

The Borel Hierarchy is a system for objectively deciding which subsets of a Polish space are superior to other ones and must be obeyed. It is defined using transfinite induction, with the base case Σ_1 = { the open sets }, Π_1 = { the closed sets }, and for a countable ordinal α, Σ_α is the collection of sets which are the countable union of sets appearing in Π_β for β < α, and dually, Π_α is the collection of sets which are the countable intersection of sets appearing in Σ_β for β < α.

The rank of a set is the index of the first time the set appears in the hierarchy. The lowest rank sets are the open and closed sets, and the highest rank set is the Borel algebra.

The Borel algebra is the σ-algebra generated by open sets, and it doesn't appear at any countable level of the Borel hierarchy; instead, it exists outside the countable hierarchy, with rank ω_1, the least uncountable ordinal, and it is a superset of all the other sets in the hierarchy.

sigma algebra

A sigma algebra (or σ-algebra) is to set algebras what sigma males are to males. The Borel hierarchy (the analog of the male hierarchy) is not a σ-algebra at any countable rank; instead the σ-algebra has the largest possible rank -- ω_1, the least uncountable ordinal (under suitable conditions e.g. the Polish space is uncountable. This is known as the non-collapse of the Borel hierarchy).

A σ-algebra is closed under countable union, countable intersection, and relative complement. This means σ-algebras arise naturally in the study of measure theory, where the measurable sets form a σ-algebra.

The algebra of Lebesgue-measurable subsets of R (i.e. the sets of real numbers with a well-defined "length", which is possibly infinity) form a sigma algebra.