Search results
Results From The WOW.Com Content Network
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
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
It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space . More precisely, if is a pre-measure defined on a ring of subsets of the space , then the set function defined by = {= |, =} is an outer measure on and the measure induced by on the -algebra of Carathéodory-measurable sets satisfies () = for (in particular, includes ).
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
Pages in category "Measure theory" The following 157 pages are in this category, out of 157 total. ... Ring of sets; Ruziewicz problem; S. Set function; Sierpiński set;
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is { 1 , 2 , 4 ...