10. Sigma-Algebras

Warning about the word “algebra”

A sigma-algebra is not an algebra over a field in the sense of the previous chapter.

Here the word “algebra” refers to a family of sets closed under set operations.

So sigma-algebras belong primarily to measure theory and probability, not to the same line as groups, rings, fields, and vector-space algebras.

Definition

Let $\Omega$ be a set.

A sigma-algebra $\mathcal F$ on $\Omega$ is a collection of subsets of $\Omega$ such that:

$\Omega\in\mathcal F$;
if $A\in\mathcal F$, then $A^c\in\mathcal F$;
if $A_1,A_2,A_3,\dots\in\mathcal F$, then

$\bigcup_{n=1}^{\infty}A_n\in\mathcal F.$ From this it follows that $\mathcal F$ is also closed under countable intersections and set differences.

Central intuition

A sigma-algebra is a language of events.

If $\Omega$ is the set of possible outcomes, then $\mathcal F$ is the collection of events about which we are allowed to ask:

Did this happen or not?

In probability, a probability measure is defined on a sigma-algebra:

$(\Omega,\mathcal F,\mathbb P).$ Here:

$\Omega$ is the space of outcomes;
$\mathcal F$ is the sigma-algebra of events;
$\mathbb P$ assigns probabilities to those events.

Why not just take all subsets?

If $\Omega$ is finite, taking all subsets is usually fine.

But for infinite spaces such as $\mathbb R$, taking all subsets can lead to pathological sets that cannot be assigned length, area, volume, or probability in a coherent way.

So measure theory chooses a sufficiently rich but controlled collection of measurable subsets.

That collection is a sigma-algebra.

Why these axioms?

Why include $\Omega$?

The certain event should be measurable.

In probability:

$\mathbb P(\Omega)=1.$

Why close under complements?

If event $A$ is meaningful, then “not $A$” should also be meaningful.

Why close under countable unions?

If $A_1,A_2,A_3,\dots$ are meaningful events, then the event

at least one of them occurs

should also be meaningful.

This is the event

$\bigcup_{n=1}^{\infty}A_n.$

Why countable operations?

The symbol $\sigma$ indicates countable closure.

Countable closure is essential because analysis and probability constantly use sequences and limits.

Events such as:

“something happens eventually”;
“something happens infinitely often”;
“a sequence converges into a set”;
“a random variable lies in some countable union of intervals”;

naturally involve countable unions and intersections.

Sigma-algebras as information

A sigma-algebra can also represent the information available to an observer.

A small sigma-algebra means the observer can distinguish only coarse events.

A large sigma-algebra means the observer can distinguish more detailed events.

Example:

Let

$\Omega=\{1,2,3,4\}.$ The collection

$\mathcal F=\{\varnothing,\Omega,\{1,2\},\{3,4\}\}$ is a sigma-algebra.

It can distinguish whether the outcome lies in ${1,2}$ or ${3,4}$, but it cannot distinguish $1$ from $2$, or $3$ from $4$.

Borel sigma-algebra

On $\mathbb R$, the most important sigma-algebra is the Borel sigma-algebra:

$\mathcal B(\mathbb R).$ It is the smallest sigma-algebra containing all open subsets of $\mathbb R$.

Intuitively, we start with open intervals and allow complements and countable unions.

This gives a huge class of measurable sets, including open sets, closed sets, intervals, countable sets, and many more.

Sigma-algebras versus topologies

A topology is closed under:

arbitrary unions;
finite intersections.

A sigma-algebra is closed under:

complements;
countable unions;
countable intersections.

Topologies are designed for continuity and local structure.

Sigma-algebras are designed for measurability and events.

Place in the build-up

Sigma-algebras are not part of the same algebraic ladder as groups, rings, fields, vector spaces, and algebras over fields.

But they share a common structural idea:

choose a class of objects and require it to be closed under the operations you need.

In groups and rings, the objects are elements.

In sigma-algebras, the objects are subsets/events.

The operations are logical set operations: not, or, and countable combinations.