Sigma-algebra generated by

See Just what is in a generated $\sigma$-algebra for examples

Theorem statement

Given a set $S\subseteq\mathcal{P}(\Omega)$ (where $\mathcal{P}(\Omega)$ denotes the power set of $\Omega$) there exists a smallest $\sigma$-algebra which we denote $\sigma(S)$ such that:

• $S\subseteq\sigma(S)$ where $$\sigma(S)=\bigcap_{\mathcal{A}\subseteq\mathcal{P}(\Omega)\text{ is a }\sigma\text{-algebra}\wedge S\subseteq\mathcal{A}}\mathcal{A}$$

We say:

• $\sigma(S)$ the $\sigma$-algebra generated by $S$
• $S$ the generator of $\sigma(S)$

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Proof:

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We will prove:

1. First that there actually is a $\sigma$-algebra that contains $S$
2. Then there is a smallest $\sigma$-algebra that contains $S$

Proof:

1. As $\mathcal{P}(\Omega)$ is a $\sigma$-algebra we know a sigma algebra containing $S$ exists.
   • So the intersection in the definition is non empty.
   • By The intersection of sets is a subset of each set we now see that $\sigma(S)\subseteq\mathcal{P}(\Omega)$
2. Using the intersection of $\sigma$-algebras is a $\sigma$-algebra (for an arbitrary indexing set)
   • If the indexing set is all the $\sigma$-algebras that contain $S$ then we see immediately that $\sigma(S)$ is the smallest $\sigma$-algebra containing $S$

The statement is proved

See also

• Ring generated by
• Borel $\sigma$-algebra