A σ-algebra is the collection of all the "events" we are allowed to assign probabilities to — the questions about a random outcome that have well-defined answers. But in practice we rarely hand over a whole σ-algebra. We start with a few sets we care about — a handful of intervals, the values a measurement can take — and ask: what is the smallest consistent event-structure that includes them?

That structure is the σ-algebra generated by a collection \mathcal{C}, written \sigma(\mathcal{C}): the smallest σ-algebra that contains \mathcal{C}. It is the minimal, fully-closed "information structure" you can build from your starting events — and, as we will see, it is exactly how probability theory pins down the idea of what is known.

"Smallest" is meant literally: \sigma(\mathcal{C}) contains \mathcal{C}, and it sits inside every other σ-algebra that contains \mathcal{C}. It is the tightest closed collection you can wrap around your wish-list.

Why a smallest one even exists

It is not obvious there is a single tightest σ-algebra rather than many incomparable ones. The guarantee comes from one clean fact: an intersection of σ-algebras is a σ-algebra. If \{\mathcal{F}_i\} are all σ-algebras on \Omega, then a set lies in \bigcap_i \mathcal{F}_i exactly when it lies in every \mathcal{F}_i — and "membership in all of them" survives complements and countable unions, because each \mathcal{F}_i is closed under them separately.

So take all the σ-algebras that contain \mathcal{C} (there is at least one — the power set 2^{\Omega}) and intersect them:

This intersection still contains \mathcal{C}, is itself a σ-algebra, and — being an intersection — is contained in each of them. That is precisely "the smallest". Notice this is an existence proof "from above": it does not tell you how to list the members of \sigma(\mathcal{C}) by building them up — which is exactly why the generated σ-algebra can be far larger than it first looks.

Worked example 1: the smallest case

Generated by a single set A \subseteq \Omega (with A neither \emptyset nor \Omega), the axioms force in exactly four sets and then stop:

We must have A; the first axiom adds \Omega; complements add A^{c} and \emptyset = \Omega^{c}; and every union of those four lands back among them — so nothing else is forced. Four sets: the finest questions you can answer knowing only whether the outcome fell in A.

Worked example 2: generated by a partition

Now suppose \{B_1, B_2, \dots, B_k\} is a finite partition of \Omega — disjoint pieces (the "atoms") whose union is all of \Omega. What is \sigma(\{B_1, \dots, B_k\})?

Closing under complements and unions, every set you can build is a union of some sub-collection of the atoms. Choosing which atoms to include is choosing a subset of \{1, \dots, k\}, so

which has exactly 2^{k} members. The single-set case is just k = 2 (the partition \{A, A^{c}\}), giving 2^{2} = 4 sets. The atoms are the smallest "resolvable" events: you can never split a B_i, only assemble unions of whole atoms.

Worked example 3: the σ-algebra generated by a random variable

The most important generated σ-algebra is the one built from a random variable. For a measurable X : \Omega \to \mathbb{R}, define

This is the collection of every event you can decide by observing the value of X alone — it captures exactly the information X reveals. Concretely, take a discrete X that takes finitely many values x_1, \dots, x_k. The level sets A_i = \{\omega : X(\omega) = x_i\} form a partition of \Omega, and by Example 2,

Knowing X tells you which atom A_i you are in — nothing finer. A constant X reveals nothing, so \sigma(X) = \{\emptyset, \Omega\}, the trivial σ-algebra.

The headline continuous example lives on the real line: the Borel σ-algebra \mathcal{B}(\mathbb{R}) = \sigma(\text{open sets}) is generated by the open sets of \mathbb{R} (equivalently, by the open intervals).