Categories: Mathematics, Measure theory.

\(\sigma\)-algebra

In set theory, given a set \(\Omega\), a \(\sigma\)-algebra is a family \(\mathcal{F}\) of subsets of \(\Omega\) with these properties:

  1. The full set is included \(\Omega \in \mathcal{F}\).
  2. For all subsets \(A\), if \(A \in \mathcal{F}\), then its complement \(\Omega \backslash A \in \mathcal{F}\) too.
  3. If two events \(A, B \in \mathcal{F}\), then their union \(A \cup B \in \mathcal{F}\) too.

This forms a Boolean algebra: property (1) represents TRUE, (2) is NOT, and (3) is AND, and that is all we need to define all logic. For example, FALSE and OR follow from the above points:

  1. The empty set is included \(\varnothing \in \mathcal{F}\).
  2. If two events \(A, B \in \mathcal{F}\), then their intersection \(A \cap B \in \mathcal{F}\) too.

For a given \(\Omega\), there are typically multiple valid \(\mathcal{F}\), in which case you need to specify your choice. Usually this would be the smallest \(\mathcal{F}\) (i.e. smallest family of subsets) that contains all subsets of special interest for the topic at hand. Likewise, a sub-\(\sigma\)-algebra is a sub-family of a certain \(\mathcal{F}\), which is a valid \(\sigma\)-algebra in its own right.

A notable \(\sigma\)-algebra is the Borel algebra \(\mathcal{B}(\Omega)\), which is defined when \(\Omega\) is a metric space, such as the real numbers \(\mathbb{R}\). Using that as an example, the Borel algebra \(\mathcal{B}(\mathbb{R})\) is defined as the family of all open intervals of the real line, and all the subsets of \(\mathbb{R}\) obtained by countable sequences of unions and intersections of those intervals. The elements of \(\mathcal{B}\) are Borel sets.

References

  1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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