σ-Algebras

The family of sets you are allowed to measure.

The idea

A $\sigma$-algebra is the family of subsets on which a measure is defined: the sets we agree to measure. Restricting to such a family is necessary because size cannot be assigned to every set. A length on the subsets of $\mathbb{R}$ that is never negative, adds over disjoint pieces, and is unchanged by sliding a set along the line cannot be defined on all subsets — sets exist so irregular that these requirements contradict one another.

The repair is not to weaken the requirements but to shrink the domain to a family closed under the operations measurement performs.

A $\sigma$-algebra $\mathcal{F}$ on a set $X$ is a collection of subsets of $X$ such that (i) $X \in \mathcal{F}$; (ii) if $A \in \mathcal{F}$ then its complement $A^{c} = X \setminus A$ is in $\mathcal{F}$; and (iii) if $A_{1}, A_{2}, A_{3}, \dots$ are all in $\mathcal{F}$, then $\bigcup_{n=1}^{\infty} A_{n}$ is in $\mathcal{F}$. The pair $(X, \mathcal{F})$ is a measurable space, and the members of $\mathcal{F}$ are the measurable sets.

Condition (iii) asks for closure under countable unions, not merely finite ones, and this is the substantial demand: a family can be closed under every finite union and still fail at an infinite one. Closure under countable unions is what lets a measure interact with limits, and the $\sigma$ in the name stands for it.

Ways to work on it

• Walkthrough. Why not every subset can be measured, the three defining conditions, countable versus finite closure, and the Borel σ-algebra.
• Proof. An intersection of σ-algebras is a σ-algebra, which is what makes the generated σ-algebra σ(E) well defined.
• Practice. Decide whether a given collection of subsets is a σ-algebra.
• Hardest. Compute a generated σ-algebra by finding the blocks its generators can distinguish.

Not sure where to start? Take the ten-question placement test.