Math 2.0

Measure and integration Definition

Pi-system

Builds on SetIntersectionSigma-algebra

Definition

Let XX be a set. A ฯ€-system on XX is a nonempty family ๐’ซ\mathcal{P} of subsets of XX such that AโˆฉBโˆˆ๐’ซA \cap B \in \mathcal{P} for all A,Bโˆˆ๐’ซA, B \in \mathcal{P}, where AโˆฉBA \cap B is the intersection.

Remarks

The family {(โˆ’โˆž,a]:aโˆˆโ„}\set{(-\infty, a] : a \in \R} is a ฯ€-system on โ„\R, since (โˆ’โˆž,a]โˆฉ(โˆ’โˆž,b]=(โˆ’โˆž,minโก(a,b)](-\infty, a] \cap (-\infty, b] = (-\infty, \min(a, b)]. The family {[0,1],[1,2]}\set{[0, 1], [1, 2]} is not a ฯ€-system, since [0,1]โˆฉ[1,2]={1}[0, 1] \cap [1, 2] = \set{1} is not in it. Every ฯƒ-algebra is a ฯ€-system. By induction, a ฯ€-system is closed under all finite intersections. The name is also written ฯ€\pi-system.

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To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionComplement, definition, no evidence levelComplementdefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionPi-system, definition, no evidence levelPi-systemdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Intersection definition
  9. Union definition
  10. Universal quantifier definition
  11. Subset definition
  12. Power set definition
  13. Universal set definition
  14. Complement definition
  15. Sigma-algebra definition
  16. Pi-system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Pi-system
About this page
TagAQN6ZN66NK
URIm2://AQN6ZN66NK
Kinddefinition
WritingW0 stub
MSC28A05, 60A10
DefinesPi-system
UsesSet, Intersection, Sigma-algebra
Changed2 November 2026

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