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Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of 𝜎-ideal is
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That is, when no 𝜎-algebra is specified, then one simply takes the full power set of the underlying set. For example, the
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of a topological space are those in the 𝜎-ideal generated by the collection of closed subsets with empty interior.
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contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being
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1079: – Set of all possible outcomes or results of a statistical trial or experiment
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1037: – Algebraic concept in measure theory, also referred to as an algebra of sets
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1052: – Family closed under complements and countable disjoint unions
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362:{\displaystyle \left\{A_{n}\right\}_{n\in \mathbb {N} }\subseteq N}
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1117:. Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
1073: – Family closed under unions and relative complements
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is a 𝜎-ideal if the following properties are satisfied:
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1067: – Family of sets closed under intersection
16:Family closed under subsets and countable unions
760:{\displaystyle x\leq y{\text{ and }}y\in I}
1085: – Algebraic structure of set algebra
1043: – Algebraic structure of set algebra
847:{\displaystyle x_{1},x_{2},\ldots \in I,}
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51:. Its most frequent application is in
47:properties. It is a special type of
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989:is a 𝜎-ideal of the power set of
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592:The notion can be generalized to
196:{\displaystyle \varnothing \in N}
1115:Measure and Integration Theory
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114:is a 𝜎-algebra of subsets of
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545:{\displaystyle S\in \Sigma }
495:{\displaystyle (X,\Sigma ),}
277:{\displaystyle B\subseteq A}
251:{\displaystyle B\in \Sigma }
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906:{\displaystyle x_{n}\leq y}
626:{\displaystyle (P,\leq ,0)}
83:{\displaystyle (X,\Sigma )}
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1050:𝜆-system (Dynkin system)
581:{\displaystyle \mu (S)=0}
1097: – Mapping function
799:(iii') given a sequence
39:(𝜎, read "sigma") is a
789:{\displaystyle x\in I,}
718:{\displaystyle 0\in I,}
167:{\displaystyle \Sigma }
107:{\displaystyle \Sigma }
43:with certain desirable
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873:{\displaystyle y\in I}
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633:with a bottom element
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515:{\displaystyle \mu }
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1056:Measurable function
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53:probability theory
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673:is a 𝜎-ideal of
666:{\displaystyle I}
646:{\displaystyle 0}
589:) is a 𝜎-ideal.
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147:{\displaystyle N}
127:{\displaystyle X}
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1095:Sigma additivity
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1035:Field of sets
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1111:Bauer, Heinz
1077:Sample space
1071:Ring of sets
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653:as follows:
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467:is given on
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502:the set of
33:sigma ideal
21:mathematics
1125:Categories
1105:References
1083:𝜎-algebra
880:such that
693:just when
552:such that
969:of a set
913:for each
898:≤
865:∈
836:∈
833:…
778:∈
752:∈
738:≤
707:∈
612:≤
594:preorders
561:μ
540:Σ
537:∈
510:μ
484:Σ
455:μ
432:countably
405:∈
385:∈
378:⋃
354:⊆
344:∈
295:∈
269:⊆
246:Σ
243:∈
217:∈
188:∈
185:∅
162:Σ
102:Σ
94:(meaning
75:Σ
37:σ-algebra
1113:(2001):
1020:See also
967:𝜎-ideal
767:implies
435:complete
284:implies
29:𝜎-ideal
1089:𝜎-ring
1065:-system
446:measure
45:closure
35:, of a
1027:δ
727:(ii')
439:filter
437:(𝜎-)
41:subset
1029:-ring
938:Thus
696:(i')
444:If a
369:then
258:then
206:When
90:be a
49:ideal
31:, or
796:and
428:dual
232:and
58:Let
27:, a
313:If
154:of
19:In
1127::
965:A
962:.
441:.
55:.
1063:π
1000:.
997:X
977:X
946:I
924:.
921:n
901:y
893:n
889:x
868:I
862:y
842:,
839:I
830:,
825:2
821:x
817:,
812:1
808:x
784:,
781:I
775:x
755:I
749:y
741:y
735:x
713:,
710:I
704:0
681:P
661:I
641:0
621:)
618:0
615:,
609:,
606:P
603:(
576:0
573:=
570:)
567:S
564:(
534:S
526:(
522:-
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481:,
478:X
475:(
411:.
408:N
400:n
396:A
389:N
382:n
357:N
348:N
341:n
336:}
331:n
327:A
323:{
310:;
298:N
292:B
272:A
266:B
240:B
220:N
214:A
203:;
191:N
142:N
122:X
78:)
72:,
69:X
66:(
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