Sigma Algebras

Why

For general measure theory, we need an algebra of sets closed under countable unions; we define such an object.¹

Definition

A countably summable subset algebra is a subset algebra for which (1) the base set is distinguished (2) the complement of a distinguished set is distinguished (3) the union of a sequence of distinguished sets is distinguished.

The name is justified, as each countably summable subset algebra is a subset algebra, because the union of $A_1, \dots , A_n$ coincides with the union of $A_1, \cdots, A_n, A_n, A_n \cdots$.

We call the set of distinguished sets a sigma algebra (or sigma field) on the base set. This language is justified (as for a regular subset algebra) by the closure properties of the sigma algebra under the usual set operations. We sometimes write are $\sigma $-algebra and $\sigma $-field.

A sub-$\sigma $-algebra (sub-sigma-algebra) is a subset of a sigma algebra which is itself a sigma algebra.

Notation

We often denote a sigma algebra by $\mathcal{A} $ or $\mathcal{F} $; the former is a mnemonic for “algebra” and the second is a mnemonic for “field”. The calligraphic typeface is meant as a reminder that the object so denoted is a set of sets.

A common pattern is also to use the calligraphic font of whichever letter is being used for the base set. Thus, if $A$ is a set, then we might choose to denote a sigma algebra on $A$ by $\mathcal{A} $.

Often, instead of saying “let $(A, \mathcal{A} )$ be a countably summable subset algebra” we say instead “let $\mathcal{A} $ be a sigma algebra on $A$.” Since the largest element of the sigma algebra is the base set, we can also say (without ambiguity): “let $\mathcal{A} $ be a sigma algebra.” The base set is $\cup \mathcal{A} $.

Examples

For any set $A$, $2^{A}$ is a sigma algebra.

For any set $A$, $\set{A, \varnothing}$ is a sigma algebra.

Let $A$ be an infinite set. Let $\mathcal{A} $ the collection of finite subsets of $A$. $\mathcal{A} $ is not a sigma algebra.

Let $A$ be an infinite set. Let $\mathcal{A} $ be the collection subsets of $A$ such that the set or its complement is finite. $\mathcal{A} $ is not a sigma algebra.

The intersection of a family of sigma algebras is a sigma algebra.

For any infinite set $A$, let $\mathcal{A} $ be the set

\[ \Set*{ B \subset A }{ \card{B} \leq \aleph_0 \lor \card{C_{A}(B)} \leq \aleph_0 }. \]

$\mathcal{A} $ is an algebra; the countable/co-countable algebra.²

Future editions will make no reference to measure theory. The entire development will follow the genetic approach, and so roughly follow the historical development for handling integration. ↩︎
Future editions will clean up and modify these examples. ↩︎