Set Rings and Algebras

Let \(X\) be some set. We say a collection of subsets \(\Sigma \subseteq X\) is a ring over \(X\) if the following conditions hold:

• \(\emptyset \in \Sigma\)
• $\A ∈ Σ \implies \(X \textbackslash A \in \Sigma\)
• \(A, B \in \Sigma \implies A\cup B \in \Sigma\)

In addition, if \(X \in \Sigma\) then \(\Sigma\) is an algebra over \(X\). Extending the finite union to a countable union leads to \(\Sigma\) being a σ-Algebra over \(X\). Similarly for rings, the resulting structure is called $σ$-rings.