Often we know which sets we want to be events — a handful of intervals, say — but that wish-list is not yet closed under the σ-algebra operations. The fix is to add exactly what the axioms force, and nothing more. The result is the σ-algebra generated by a collection \mathcal{C}, written \sigma(\mathcal{C}): the smallest σ-algebra that contains \mathcal{C}.

"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".

The smallest example

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.

The headline 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}.