Borel Set - Generating The Borel Algebra

Generating The Borel Algebra

In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows.

For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let

• be all countable unions of elements of T
• be all countable intersections of elements of T

Now define by transfinite induction a sequence Gm, where m is an ordinal number, in the following manner:

• For the base case of the definition, let be the collection of open subsets of X.
• If i is not a limit ordinal, then i has an immediately preceding ordinal i − 1. Let

• If i is a limit ordinal, set

We now claim that the Borel algebra is Gω₁, where ω₁ is the first uncountable ordinal number. That is, the Borel algebra can be generated from the class of open sets by iterating the operation

to the first uncountable ordinal.

To prove this fact, note that any open set in a metric space is the union of an increasing sequence of closed sets. In particular, it is easy to show that complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountable limit ordinal, Gm is closed under countable unions.

Note that for each Borel set B, there is some countable ordinal α_B such that B can be obtained by iterating the operation over α_B. However, as B varies over all Borel sets, α_B will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω₁, the first uncountable ordinal.