Sigma-algebra

A sigma-algebra on $\Omega$ contains $\Omega$ and is closed under complements and countable unions.

A nonempty collection of subsets of a fixed universe closed under complements and finite unions, hence also finite intersections. It generates a sigma-algebra by allowing countable set operations. A countable generating algebra provides one simultaneous family of test indicators for measure-theoretic arguments.

A countable algebra of sets generating the intended sigma-algebra. Standard Borel spaces admit such algebras by taking finite Boolean combinations from a countable topological base. Finite linear combinations of its indicators are dense in $L^2$ for every Borel probability measure, by the Monotone class theorem. Countability permits a common full-measure convergence set for all test indicators.

A subset of $X$ is measurable relative to a sigma-algebra $\mathcal{F}$ when it belongs to $\mathcal{F}$. A measure defined on $\mathcal{F}$ assigns sizes to these sets. Measurability depends on the chosen sigma-algebra; for example, a Lebesgue measurable set need not be a Borel set.

An atom of a sigma-algebra is a nonempty measurable set containing no proper nonempty measurable subset. A finite sigma-algebra partitions its underlying set into atoms, and each measurable set is a union of them.

A Borel set in a topological space belongs to the smallest sigma-algebra containing every open set.

The Borel sigma-algebra of a topological space $X$ is the sigma-algebra generated by its open sets. Its members are exactly the Borel sets.

Every subset of a discrete space is open, so its Borel sigma-algebra is the full power set.

A pi-system is a nonempty family of subsets closed under finite intersections.

A Dynkin system contains $\Omega$, is closed under complements, and is closed under countable disjoint unions. Equivalently, it contains differences $B\setminus A$ whenever $A\subseteq B$ are members.

Every Dynkin system containing a pi-system $\mathcal{P}$ also contains the sigma-algebra $\sigma (\mathcal{P})$. Equivalently, the Dynkin system generated by a pi-system equals the sigma-algebra it generates.