Borel Algebra

Definition

The most important set of \fs4 \sigma-algebras is the Borel (\fs4 \sigma-)algebra which is defined on the set of real numbers (or more generally on any Euclidean space like \fs4 \mathbb{R}^n).

This is the smallest \fs4 \sigma-algebra that contains all open sets of \fs4 \mathbb{R}. Loosely speaking the Borel algebra is generated by all open sets of \fs4 \mathbb{R}, by taking all possible (countable) unions, intersections and complements. The members of the Borel algebra are called Borel sets.

We denote the Borel algebra with \fs4 \mathscr{B}=\mathscr{B}(\mathbb{R}).

Examples

For example the sets

  1. \fs4 (-\pi, 7) \in \mathscr{B}
  2. \fs4 \{-\pi\} = (-\infty, -\pi)^C \cap (-\pi, \infty)^C \in \mathscr{B}
  3. \fs4 [-\pi, 7) = \{-\pi\} \cup (-\pi, 7) \in \mathscr{B}
  4. \fs4 \mathbb{Q} = \cup \left\{ q: q \in \mathbb{Q} \right\} \in \mathcal{B} since rationals are countable

Virtually any sets that we can imagine will be elements of the Borel algebra, but there are some weird constructs that are not. An example of a set that is not a Borel set is the Vitali set, \fs4 V \notin \mathscr{B}.

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