Tag Archives: Generated sigma algebra

Clarification on the σ-algebra generated by a set

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}).