Sigma-algebra

A sigma-algebra on a set is a collection of subsets containing the empty set and closed under complements and countable unions. It specifies which subsets are measurable.

Power set: The set of all subsets of a given set. A sigma-algebra is a selected subcollection of this larger family.
Measurable space: A set equipped with a sigma-algebra of its measurable subsets. It packages the underlying set together with the sigma-algebra being studied.
Generated sigma-algebra: The smallest sigma-algebra containing a specified collection of subsets. It turns any starting family of sets into the least sigma-algebra that contains it.
Trivial sigma-algebra: The sigma-algebra consisting only of the empty set and the whole space. It is the smallest possible sigma-algebra and distinguishes no proper subsets.
Complement (set theory): The elements of a universal set that are not in a specified subset. Closure under complements is one of the defining requirements.
Probability space: A measurable space equipped with a probability measure assigning total mass one. Its events are precisely the sets in the space's sigma-algebra.
Monotone class theorem: A theorem connecting closure under monotone limits with the sigma-algebra generated by a set family. It provides a powerful way to prove identities throughout a generated sigma-algebra.
Discrete sigma-algebra: The power set of a set, treated as a sigma-algebra. It is the largest sigma-algebra on the space, making every subset measurable.
Countable union: The union of a finite or countably infinite family of sets. A sigma-algebra must contain unions formed from countably many of its sets.
Random variable: A measurable function from a probability space into a measurable target space. Measurability is defined by how inverse images interact with sigma-algebras.

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