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In measure theory, a nonempty family of sets is called a ring (of sets) if it is closed under union and relative complement (set-theoretic difference). [2] That is, the following two statements are true for all sets A {\displaystyle A} and B {\displaystyle B} ,
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite.
δ-ring – Ring closed under countable intersections Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets Join (sigma algebra) – Algebraic structure of set algebra Pages displaying short descriptions of redirect targets
For example, the symmetric difference of the sets ... This is the prototypical example of a Boolean ring. ... Using the ideas of measure theory, ...
A set function generally aims to measure subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members; Combinatorial design – Symmetric arrangement of finite sets; δ-ring – Ring closed under countable intersections; Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
δ-ring – Ring closed under countable intersections; Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets; Ideal (set theory) – Non-empty family of sets that is closed under finite unions and subsets
Littlewood stated the principles in his 1944 Lectures on the Theory of Functions [1] as: . There are three principles, roughly expressible in the following terms: Every set is nearly a finite sum of intervals; every function (of class L p) is nearly continuous; every convergent sequence of functions is nearly uniformly convergent.