Generated Sigma Algebras
Why
A simple (albeit indirect) way to obtain a sigma algebra, is to start with some sets, and then to add all the sets needed to make the starting set closed under the various operations.
Definition
The generated sigma algebra for a set of subsets is the smallest sigma algbera containing the set of subsets. We must prove the existence and uniqueness of this sigma algebra.
- For all $\alpha \in I$, $A \in \mathcal{A} _{\alpha }$, thus $A \in \mathcal{A} $; condition (a).
- For all $B \in \mathcal{A} $, for all $\alpha \in I$, $B \in \mathcal{A} _{\alpha }$. Thus, for all $\alpha \in I$, $C_{A}(B) \in \mathcal{A} _{\alpha }$. And so $C_{A}(B) \in \mathcal{A} $; condition (b).
- For all sequences $\set{B_n} \subset \mathcal{A} $, $\set{B_n} \subset \mathcal{A} _{\alpha }$ for all $\alpha $. Thus $\cup_{n} B_n \in \mathcal{A} _{\alpha }$ for all $\alpha $ and so $\cup_{n} B_n \in \mathcal{A} $; condition (c).
On the other hand, the union of a set of sigma algebras can fail to be a sigma algebra.
We know of one sigma algebra containing $\mathcal{A} $: the power set of $A$. Thus, the set of sigma algebras containing $\mathcal{A} $ is not empty. Proposition~\ref{intersectsigmaalgebras} implies the intersection of all such sigma algebras (containing $\mathcal{A} $) is a sigma algebra. The intersection contains $\mathcal{A} $, and is contained in all other sigma algebras with this property, so is a smallest sigma algebra containing $\mathcal{A} $. If $\mathcal{B} , \mathcal{C} $ were two smallest sigma algebras, then $\mathcal{B} \subset \mathcal{C} $ and $\mathcal{C} \subset \mathcal{B} $, but then $\mathcal{B} = \mathcal{C} $; thus the smallest sigma algebra is unique.
Notation
Let $A$ be a set and $\mathcal{A} \subset \powerset{A}$. We denote the sigma algebra generated by $\mathcal{A} $ by $\sigma (\mathcal{A} )$.