Consider a set \(\Omega\) and a collection \(\mathcal{A}\) of subsets of \(\Omega\). We denote all [[sigma-algebra|\(\sigma\)-algebra]] containing \(\mathcal{A}\) by \(\mathcal{F}_{i},~i \in I\) where \(I\) is some index set. There exists a smallest \(\sigma\)-algebra containing \(\mathcal{A}\) denoted \(\sigma(\mathcal{A})\) - known as the [[generated-sigma-algebra|\(\sigma\)-algebra generated by \(\mathcal{A}\)]] - which is the intersection of all other \(\sigma\)-algebra containing \(\mathcal{A}\), i.e. \[ \sigma(\mathcal{A}) := \bigcap_{i\in I}\mathcal{F}_{i}. \]
\begin{proof} To show that \(\sigma(\mathcal{A})\) is a \(\sigma\)-algebra we simply note each of the following: 1. \(\forall i,\Omega \in\mathcal{F}_{i} \implies\Omega \in\cap_{i}\mathcal{F}_{i}\). 2. Let \(A\in\cap_{i}\mathcal{F}_{i}\implies \forall i, A\in\mathcal{F}_{i}\implies \forall i,A^c\in\mathcal{F}_{i}\implies A^c\in\cap_{i}\mathcal{F}_{i}\). 3. Let \(\{ A_{n} \}_{n\in\mathbb{N}}\subseteq \cap_{i}\mathcal{F}_{i}\implies \{ A_{n} \}_{n\in \mathbb{N}}\in \mathcal{F}_{i},~\forall i\in I\implies \bigcup_{n\in \mathbb{N}}A_{n}\in \mathcal{F}_{i},~ \forall i\in I\implies \bigcup_{n\in \mathbb{N}}A_{n}\in \bigcap_{i \in I}\mathcal{F}_{i}\). Hence \(\sigma(\mathcal{A})\) is a \(\sigma\)-algebra and since it is the intersection of all possible \(\sigma\)-algebra containing \(\mathcal{A}\) it is the smallest.\end{proof}