Measure Theoretic Probability - Video 8 - Sigma Algebras
Sigma Algebras
A \(\sigma\)-algebra, call it \(\mathcal S\), is an algebra that is closed under countable unions. Closure under countable unions means that if \(E_1,E_2,\dots \in \mathcal S\) then \(\bigcup_{i\ge 1}E_i \in\mathcal S\).
No Need to Check Pairwise Unions
When given a collection \(\mathcal S\) that you want to check for being a \(\sigma\)-algebra, there is no need to check for closure under pairwise unions. This is because if any collection contains the universe, is closed under complements, and is closed under countable unions, then it must necessarily be closed under pairwise unions.
Proof: Let \(\mathcal S\subseteq \mathcal P(\Omega)\) be a collection of subsets of \(\Omega\) which contains the universe, is closed under complements, and closed under countable unions. Then \(\Omega^c=\emptyset\in\mathcal S\) by containing the universe and closure under complements. Then for any \(E_1,E_2\in\mathcal S\) we may construct the sequence of sets \(D_1=E_1, D_2=E_2, D_n = \emptyset\) for all \( n\ge 2\). Then by closure under countable unions,
\[E_1\cup E_2 = \bigcup_{i\ge 1} D_i \in \mathcal S \]
which proves closure under pairwise unions.