AKA σ-algebra, σ-field, 시그마-대수, 시그마-체
Twins:
https://mathworld.wolfram.com/BorelSigma-Algebra.html
https://proofwiki.org/wiki/Definition:Borel_Sigma-Algebra
https://freshrimpsushi.github.io/posts/borel-sigma-algebra-borel-measurable-space/
https://mathworld.wolfram.com/BorelSigma-Algebra.html
https://proofwiki.org/wiki/Definition:Borel_Sigma-Algebra
https://freshrimpsushi.github.io/posts/borel-sigma-algebra-borel-measurable-space/
Up: 대수학,algebra > 보렐_대수,Borel_algebra and 시그마대수,sigma-algebra ?
}
(Borel_sigma-algebra를 줄여 Borel_algebra 간단히 부르기도 함. - freshrimp) - QQQ 동의어인지 chk
borel sigma algebra}
cylindrical_sigma-algebra w
{
cylindrical σ-algebra
AKA
product σ-algebra
MKL
cylinder_set w
...cylindrical σ-algebra product σ-algebra
}
cylindrical σ-algebra
AKA
product σ-algebra
MKL
cylinder_set w
{
cylinder set
원기둥집합 ?
Sub:
cylinder_set_measure w
cylinder set 보면 통집합 원주집합 원기둥집합 등 통일 안 되어 있음. wk는 기둥_집합 ... 이대로라면 원통집합 도 가능?
... cylinder set
}
Cylindrical_σ-algebracylinder set
원기둥집합 ?
Sub:
cylinder_set_measure w
cylinder set 보면 통집합 원주집합 원기둥집합 등 통일 안 되어 있음. wk는 기둥_집합 ... 이대로라면 원통집합 도 가능?
... cylinder set
}
...cylindrical σ-algebra product σ-algebra
}
// 수백
//mw
집합,set 가 있으면, σ-algebra 는 의 부분집합,subsets들의 nonempty collection(하나 이상의 모음?) 이다. 다음 세 조건을 만족하는.
// F가 set이란 말이 없고 collection이라 해서 집합기호를 쓰지 않음. 나중에 정확히 reread
를 포함한 모든 σ-algebra들의 교집합,intersection을 take하면, we obtain the smallest such σ-algebra. //일단 번역 않고 놔둠.
'Smallest σ-algebra containing S'를, 'σ-algebra generated by S'라고 부른다.
// F가 set이란 말이 없고 collection이라 해서 집합기호를 쓰지 않음. 나중에 정확히 reread
- is in
- If is in then is in
- 만약 이 의 원소,element들의 수열,sequence이면, 의 합집합,union is in
를 포함한 모든 σ-algebra들의 교집합,intersection을 take하면, we obtain the smallest such σ-algebra. //일단 번역 않고 놔둠.
'Smallest σ-algebra containing S'를, 'σ-algebra generated by S'라고 부른다.
mklink
집합,set 체,field랑 관련이 있는듯. 이름조차도 AKA sigma-field
특히 Borel_field, Borel_set - curr see 체,field#s-3
집합,set 체,field랑 관련이 있는듯. 이름조차도 AKA sigma-field
특히 Borel_field, Borel_set - curr see 체,field#s-3
측도,measure...
AKA Boolean σ-algebra. 불_대수,Boolean_algebra에서 특정 조건을 만족하는.
// (아주 대충) 네가지임. 정확히 rewrite.
- 자기자신을 포함
- closed under complement
- closed under union
- closed under intersection
tmp from http://contents2.kocw.or.kr/KOCW/html/2017/kumoh/limwansu091/1/default.htm 5:30
{
Field (ℱ) : a set of subsets of Ω
{
Field (ℱ) : a set of subsets of Ω
ℱ is assumed to be a σ-algebra, meaning it satisfies the following axioms
- Ω ∈ ℱ
- If A ∈ ℱ then AC ∈ ℱ
- If A, B ∈ ℱ then A ∪ B ∈ ℱ. Also, if A1, A2, … is a sequence of elements in ℱ then
1. Sub: Borel sigma algebra ¶
보렐_대수,Borel_algebra : 열린집합,open_set과 닫힌집합,closed_set을 모두 포함하는 가장 작은 시그마대수,sigma-algebra
보렐_대수 Borel_algebra = 보렐_시그마대수 Borel_sigma-algebra 인듯.....? chk
보렐_대수 Borel_algebra = 보렐_시그마대수 Borel_sigma-algebra 인듯.....? chk
// tmp from 수학백과: 시그마 대수 section 3.
{
"위상공간,topological_space에서 모든 열린집합,open_set을 포함하는 가장 작은 시그마 대수를 보렐 시그마 대수(Borel σ-algebra)라고 한다. 보렐 시그마 대수의 원소를 보렐_집합,Borel_set이라고 한다."
}
{
"위상공간,topological_space에서 모든 열린집합,open_set을 포함하는 가장 작은 시그마 대수를 보렐 시그마 대수(Borel σ-algebra)라고 한다. 보렐 시그마 대수의 원소를 보렐_집합,Borel_set이라고 한다."
}
3. 이름의 sigma에 대한 설명: via 모형_이론#보편_대수학 at 2023-10-13 ¶
{
분야는
(모형이론 모델이론 모형론 모델론 model_theory model_theory ) > ( 보편대수 보편대수학 universal_algebra 보편_대수학 universal_algebra Universal_algebra x(2023-10에 없음, 2007년 지워짐) Universal_algebra )관련임.
분야는
(모형이론 모델이론 모형론 모델론 model_theory model_theory ) > ( 보편대수 보편대수학 universal_algebra 보편_대수학 universal_algebra Universal_algebra x(2023-10에 없음, 2007년 지워짐) Universal_algebra )관련임.
"보편 대수학의 주요 개념은 부호수(signature) σ와 그것에 의해 규정되는 구조인 σ-대수(σ-algebra)이다. 예시는...."
so rel:
구조,structure
esp 수학구조/수학적구조 mathematical_structure ( mathematical_structure Mathematical_structure 수학적_구조 数学的構造 ... 수학 구조 )
부호수,signature? signature 부호수
구조,structure
esp 수학구조/수학적구조 mathematical_structure ( mathematical_structure Mathematical_structure 수학적_구조 数学的構造 ... 수학 구조 )
부호수,signature? signature 부호수
즉 structure에는 signature가 붙을 수 있고 그 중 하나가 'sigma'? 아님 혹시 sigma가 signature의 앞글자를 딴 건지? CHK
4. tmp bmks ko ¶
https://jrc-park.tistory.com/290?category=487573
https://hanmaths.tistory.com/36 - 정의
https://hanmaths.tistory.com/40 - 필요충분조건
https://hanmaths.tistory.com/42 - 예
측도#s-2.1 - 측도,measure를 설명하기 직전에 정의함.
https://elementary-physics.tistory.com/123?category=860181 - 마지막부분 Advanced Topics
https://blog.naver.com/mykepzzang/222197904497 - σ-(시그마-)로 이름붙여진 게 무슨 뜻인지 설명
집합 X에 대한 σ-algebra는, X의 부분집합들의 collection Σ이며 다음 네 조건을 만족하는 것.
- X 자신을 포함
- closed under complement (여집합,complement 연산에 닫혀있음)
- closed under union (셀 수 있는 횟수의 합집합,union 연산에 닫혀있음?)
- closed under countable intersection (셀 수 있는 횟수의 교집합,intersection 연산에 닫혀있음?)
rel. 닫힘성,closedness
https://hanmaths.tistory.com/36 - 정의
https://hanmaths.tistory.com/40 - 필요충분조건
https://hanmaths.tistory.com/42 - 예
측도#s-2.1 - 측도,measure를 설명하기 직전에 정의함.
https://elementary-physics.tistory.com/123?category=860181 - 마지막부분 Advanced Topics
https://blog.naver.com/mykepzzang/222197904497 - σ-(시그마-)로 이름붙여진 게 무슨 뜻인지 설명
6. Bmks en ¶
Mathonline: Measure Theory (측도론, measure_theory, 측도,measure 이론 설명)에서 > 2. Algebras of Sets > σ-Algebras of Sets 참조
http://mathonline.wikidot.com/measure-theory
http://mathonline.wikidot.com/measure-theory
Twins:
수학백과: 시그마 대수
https://mathworld.wolfram.com/Sigma-Algebra.html
https://proofwiki.org/wiki/Definition:Sigma-Algebra
σ-algebra
시그마_대수
https://encyclopediaofmath.org/wiki/Algebra_of_sets (2.) (https://encyclopediaofmath.org/wiki/Sigma-algebra 는 Algebra of sets로 redirect.)
https://everything2.com/title/%26sigma%3B algebra
完全加法族 - 일본어는 '완전가법족'으로 번역.
https://ncatlab.org/nlab/show/sigma-algebra
수학백과: 시그마 대수
https://mathworld.wolfram.com/Sigma-Algebra.html
https://proofwiki.org/wiki/Definition:Sigma-Algebra
σ-algebra
시그마_대수
https://encyclopediaofmath.org/wiki/Algebra_of_sets (2.) (https://encyclopediaofmath.org/wiki/Sigma-algebra 는 Algebra of sets로 redirect.)
https://everything2.com/title/%26sigma%3B algebra
完全加法族 - 일본어는 '완전가법족'으로 번역.
https://ncatlab.org/nlab/show/sigma-algebra
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