The previous page ended with a defeat that turns out to be a beginning. Vitali's set proved that no translation-invariant, countably additive length can be defined on every subset of \mathbb{R}. Something has to give, and the honest thing to give up is totality: we stop insisting that a measure be defined everywhere, and instead pick a rich, well-behaved family of sets — the measurable ones — on which everything works.

But "well-behaved" cannot mean an arbitrary collection. Analysis is built on operations: complements ("not in A"), unions ("in A or B"), and — because we chase limits everywhere — countable unions (\bigcup_n A_n, the set-theoretic shadow of a limit). If we can measure A, we had better be able to measure its complement and its countable combinations, or the theory falls apart the moment we take a limit. A family closed under exactly these operations is a σ-algebra, and it is the natural home — the domain — of every measure. This page is about that structure alone: what it is, what it forces, and where the useful ones come from.

The definition — three axioms

Let X be a set (the "universe"). A collection \mathcal{F} of subsets of X — a subfamily of the power set \mathcal{P}(X) — is a σ-algebra on X if it satisfies three closure conditions:

• (i) Contains the whole space: X \in \mathcal{F}.
• (ii) Closed under complement: if A \in \mathcal{F} then A^c = X \setminus A \in \mathcal{F}.
• (iii) Closed under countable unions: if A_1, A_2, A_3, \dots \in \mathcal{F} then \displaystyle\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}.

That is the entire definition. Three axioms — and yet they carry a surprising amount of freight. Everything else you want is forced:

• The empty set belongs: \varnothing = X^c, so \varnothing \in \mathcal{F} by (i) and (ii).
• Closed under countable intersections: by De Morgan, \bigcap_n A_n = \bigl(\bigcup_n A_n^c\bigr)^c — complement each set (ii), take the countable union (iii), complement again (ii). All three steps stay inside \mathcal{F}.
• Closed under finite unions and intersections: a finite union is a countable one with the tail padded by \varnothing: A \cup B = A \cup B \cup \varnothing \cup \varnothing \cup \cdots.
• Closed under set differences and symmetric differences: A \setminus B = A \cap B^c and A \,\triangle\, B = (A\setminus B)\cup(B\setminus A) are built from the operations above.

So a σ-algebra is a family of sets on which the whole Boolean algebra of set operations, extended to countably many arguments, can be performed without ever leaving the family. It is algebraically self-contained — which is exactly what a measure needs beneath it.

The "σ" is the whole point: algebra vs σ-algebra

Drop condition (iii) down to finite unions and you get the weaker notion of an algebra (or "field") of sets: closed under complement and under A \cup B for two sets — hence any finite union. The prefix σ (for the German Summe, countable sum) is precisely the upgrade from finite to countable. Every σ-algebra is an algebra; the converse fails.

Why does this one word matter so much? Because measure theory lives on limits, and a limit of sets is a countable operation. Monotone limits \bigcup_{n} A_n and \bigcap_n A_n, the \limsup and \liminf of a sequence of events, the super-level sets that appear when you integrate — all of them assemble countably many pieces. A merely finite algebra would let you take two steps toward a limit and then strand you. Concretely:

• Let \mathcal{A} be the collection of subsets of \mathbb{N} that are finite or cofinite (complement finite). This is an algebra: it is closed under complement and finite unions. But it is not a σ-algebra — the singletons \{2\}, \{4\}, \{6\}, \dots all lie in \mathcal{A}, yet their countable union, the even numbers, is neither finite nor cofinite, so it escapes \mathcal{A}.

That one example is the whole moral in miniature: finite closure is not enough to survive a limit; countable closure is exactly enough.

A gallery of σ-algebras

Four constructions cover almost everything you meet, from the trivial to the indispensable.

1. The trivial σ-algebra

\mathcal{F} = \{\varnothing, X\}. The smallest σ-algebra on any X — it contains only "nothing" and "everything". Check: X is in, complements swap the two members, and every union is one of them. It is measure theory's featureless vacuum: measurable, but it cannot distinguish any two points.

2. The power set

\mathcal{F} = \mathcal{P}(X), all subsets. The largest σ-algebra on X — trivially closed under everything. For a finite X with |X| = n this has 2^n members, and it is the sensible choice for discrete and countable universes (a coin, a die, \mathbb{N}). It is only on uncountable spaces like \mathbb{R} that \mathcal{P}(X) becomes too big to carry Lebesgue measure — the lesson of Vitali — and we are forced to a smaller σ-algebra.

3. The countable / co-countable σ-algebra

On any X, let \mathcal{F} = \{\, A \subseteq X : A \text{ is countable or } A^c \text{ is countable}\,\}. This is a σ-algebra (unlike the finite/cofinite algebra above): X is co-countable (its complement \varnothing is countable); complements are handled by the symmetric "or"; and a countable union of members is countable if all are countable, and otherwise has a co-countable member forcing the union co-countable. The countability of a countable union of countable sets is exactly what makes (iii) hold — and exactly what failed for the finite/cofinite algebra. It is a genuinely useful test case and a favourite exam trap.

4. The σ-algebra generated by a collection

Given any family \mathcal{E} \subseteq \mathcal{P}(X) of sets you care about, there is a smallest σ-algebra containing it, written \sigma(\mathcal{E}) and called the σ-algebra generated by \mathcal{E}. The construction is a classic piece of "intersect-everything" abstract nonsense:

The intersection is non-empty as a construction because \mathcal{P}(X) is always one such \mathcal{G}. The one thing to verify is that an intersection of σ-algebras is again a σ-algebra — and it is, axiom by axiom: if every \mathcal{G} contains X, so does the intersection; if A lies in every \mathcal{G} then so does A^c, hence in the intersection; and if each A_n lies in every \mathcal{G}, then \bigcup_n A_n does too. So \sigma(\mathcal{E}) is a σ-algebra, it contains \mathcal{E}, and being an intersection it is contained in every σ-algebra that contains \mathcal{E} — i.e. it is the smallest one.

The star of the show: the Borel σ-algebra

On \mathbb{R} the sets we actually want to measure are built from intervals. So take \mathcal{E} to be the open intervals — or equivalently all open sets — and generate. The result is the Borel σ-algebra, Émile Borel's creation:

The two descriptions agree because every open subset of \mathbb{R} is a countable union of open intervals, so each generating family sits inside the σ-algebra generated by the other. Because \mathcal{B}(\mathbb{R}) is closed under complement and countable unions and intersections, it is enormous — it contains, at a minimum:

• every open set and (by complement) every closed set;
• every half-open and closed interval, and every singleton \{a\} = \bigcap_n (a - \tfrac1n,\, a + \tfrac1n);
• every countable set — so \mathbb{Q} and hence the irrationals are Borel;
• every F_\sigma (countable union of closed sets) and G_\delta (countable intersection of open sets), and the whole transfinite tower built by iterating these.

Yet \mathcal{B}(\mathbb{R}) is not everything. A counting argument shows there are only \mathfrak{c} = 2^{\aleph_0} Borel sets but 2^{\mathfrak{c}} subsets of \mathbb{R} in total, so most subsets are not Borel. And there is a strict tower on the way up:

where \mathcal{L}(\mathbb{R}) is the (larger) σ-algebra of Lebesgue-measurable sets — Borel sets completed by all subsets of measure-zero sets. Both inclusions are strict: there are Lebesgue-measurable sets that are not Borel, and — by Vitali — subsets of \mathbb{R} that are not Lebesgue-measurable at all. The picture below is that hierarchy.