Generated sigma algebra | "There is no royal road to geometry". Euclid

This post elaborates on the concept of σ-algebra generated by a set, as presented in p. 17 of the book “Probability with martingales” by David Williams.

Let $\mathcal{C}$ be a class of subsets of a set $S$ . It is stated in the book that the σ-algebra $\sigma(\mathcal{C})$ generated by $\mathcal{C}$ is the smallest σ-algebra on $S$ such that contains $\mathcal{C}$ . Moreover, it is mentioned that $\sigma(\mathcal{C})$ is the intersection of all σ-algebras on $S$ that contain $\mathcal{C}$ .

Three clarifications will be made. Let $\mathcal{G}:=\{\sigma\mbox{-algebra }\Sigma_i\mbox{ on }S\mbox{ with }\mathcal{C}\subseteq \Sigma_i:i\in I\}$ be the set of all σ-algebras on $S$ that contain $\mathcal{C}$ .

Firstly, it will be shown that the intersection $\underset{i}{\bigcap}\Sigma_i$ , which is the intersection of all σ-algebras on $S$ that contain $\mathcal{C}$ , is itself a σ-algebra on $S$ that contains $\mathcal{C}$ . Since $S\in\Sigma_i$ for all $i\in I$ , it follows that $S\in\underset{i}{\bigcap}\Sigma_i$ . Moreover, consider a set $F\in\underset{i}{\bigcap}\Sigma_i$ . So $F\in\Sigma_i$ for all $i$ , which means $F^{c}\in\Sigma_i$ for all $i$ , so $F^{c}\in\underset{i}{\bigcap}\Sigma_i$ . Let $\{F_j:j\in J\}$ be a countable collection of sets $F_j\in\underset{i}{\bigcap}\Sigma_i$ . It follows that $F_j\in \Sigma_i$ for all $i\in I$ and all $j\in J$ , so $\underset{j}{\bigcup}F_j\in\Sigma_i$ for all $i$ , which gives $\underset{j}{\bigcup}F_j\in\underset{i}{\bigcap}\Sigma_i$ . It has thus been confirmed that $\underset{i}{\bigcap}\Sigma_i$ is a σ-algebra on $S$ . As $\mathcal{C}\subseteq\Sigma_i$ for all $i$ , it becomes obvious that $\mathcal{C}\subseteq\underset{i}{\bigcap}\Sigma_i$ , so the σ-algebra $\underset{i}{\bigcap}\Sigma_i$ on $S$ contains $\mathcal{C}$ .

Consider now those σ-algebras $\Sigma_k\in\mathcal{G}$ with $\mathcal{C}\subseteq\Sigma_k$ that satisfy $\Sigma_k\subseteq\Sigma_i$ for all $\Sigma_i\in\mathcal{G}$ with $\mathcal{C}\subseteq\Sigma_i$ . This is what is meant by “smallest σ-algebra $\Sigma$ on $S$ such that $\mathcal{C}\subseteq\Sigma$ ” in William’s book. Assume that there are two such distinct σ-algebras $\Sigma_k$ and $\Sigma_{\ell}$ , i.e. $\Sigma_k\neq\Sigma_{\ell}$ . Since $\Sigma_{\ell}\in\mathcal{G}$ , it is then deduced that $\Sigma_k\subseteq\Sigma_{\ell}$ . Similarly, $\Sigma_{\ell}\subseteq\Sigma_{k}$ , so $\Sigma_k=\Sigma_{\ell}$ , which is a contradiction. Hence, there is a single σ-algebra in $\mathcal{G}$ that contains $\mathcal{C}$ and is a subset of any other σ-algebra in $\mathcal{G}$ that also contains $\mathcal{C}$ .

It has been shown that $\underset{i}{\bigcap}\Sigma_i\in\mathcal{G}$ . Moreover, $\underset{i}{\bigcap}\Sigma_i$ satisfies $\underset{i}{\bigcap}\Sigma_i\subseteq\Sigma$ for any $\Sigma\in\mathcal{G}$ . Uniqueness has also been established, so $\underset{i}{\bigcap}\Sigma_i$ is the only element in $\mathcal{G}$ that is a subset of all $\Sigma\in\mathcal{G}$ . Put in words, $\underset{i}{\bigcap}\Sigma_i$ is the only σ-algebra on $S$ that contains $\mathcal{C}$ and is contained in any other σ-algebra $\Sigma$ on $S$ that satisfies $\mathcal{C}\subseteq\Sigma$ .

$\underset{i}{\bigcap}\Sigma_i$ is called the σ-algebra generated by $\mathcal{C}$ and is denoted by $\sigma(\mathcal{C})$ .

"There is no royal road to geometry". Euclid

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Clarification on the σ-algebra generated by a set