Set Algebras

Published

28 June 2024

Here, we collect some definitions regarding collections of sets and their algebraic structures.

Let $X$ be a nonempty set. An algebra of sets on $X$ is a nonempty collection $\mathcal{A}$ of subsets of $X$, $\mathcal{A} \subset 2^X$, such that
  1. $\mathcal{A}$ is closed under complements
  2. $\mathcal{A}$ is closed under finite unions.
A $\sigma$-algebra on $X$ is an algebra that is closed under countable unions.

Some easy properties of algebras are as follows. We’ll skip the proof since it is a straightforward consequence of the definitions.

Let $\mathcal{A}$ be an algebra (resp. $\sigma$-algebra). Then,
  1. $\varnothing \in \mathcal{A}$ and $X \in \mathcal{A}$.
  2. $\mathcal{A}$ is closed under finite (resp. countable) intersections.
  3. If $\mathcal{A}$ is closed under countable disjoint unions, it is a $\sigma$-algebra.
  4. The intersection of any collection of $\sigma$-algebras on $X$ is a $\sigma$-algebra on $X$.

References.

  1. Folland, G. B. (1999). Real analysis: Modern techniques and their applications (2nd ed.).