Borel measure

A measure defined on the Borel sets of a topological space, the sigma-algebra generated by its open sets. It assigns sizes such as length, area, or probability while respecting countable additivity.

Borel sigma-algebra: The smallest sigma-algebra containing every open set of a topological space. It is the precise collection of sets on which a Borel measure is defined.
Carathéodory extension theorem: A theorem that extends a premeasure on an algebra to a measure on the sigma-algebra it generates, under suitable conditions. It provides a standard route from simple set functions to Borel measures.
Probability measure: A measure assigning total mass one to a measurable space. When the Borel space has total mass one, its Borel measure models a random outcome.
Lebesgue-Stieltjes measure: A measure on the real line associated with a nondecreasing right-continuous function. It shows how distribution functions can specify Borel measures on the real line.
Topological space: A set equipped with a collection of open subsets satisfying axioms for unions and finite intersections. Its open sets determine the Borel sets available to the measure.
Outer measure: A countably subadditive set function defined on all subsets of a space. Carathéodory's method uses outer measures to identify measurable Borel sets.
Probability space: A measurable space equipped with a probability measure. Borel probability measures provide the event probabilities in many topological models of randomness.
Counting measure: A measure assigning each set its number of elements, possibly infinite. On a suitable space, counting measure is a Borel measure with a very different notion of size.
Sigma-algebra: A collection of subsets containing the whole space and closed under complements and countable unions. The Borel sets form a sigma-algebra, which makes countable measure operations well-defined.
Regular measure: A measure approximable from outside by open sets and, in suitable settings, from inside by compact sets. Regularity links a Borel measure's values to the topology's open and compact sets.

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