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| = σ Algebra Notation = | = σ Algebra = |
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| σ algebra uses and re-uses many common notations. See also some [[Statistics/ProbabilityNotation|probability notation]], [[Statistics/BayesianNotation|Bayesian notation]], [[Statistics/JointProbability|joint probability notation]], [[Statistics/ConditionalProbability|conditional probability notation]], [[Statistics/ExpectedValues|expected value notation]], and [[Statistics/ConditionalExpectations|conditional expectation notation]]. |
A '''σ algebra''' (sigma algebra) is a measurable set. |
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| == Sets and Subsets == | == Description == |
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| The maximal set, which in probability applications is the '''sample space''', is notated as ''Ω''. | A σ algebra is usually notated using a calligraphic uppercase letters, like 𝒜. |
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| For a given sample space ''Ω'', a σ algebra 𝒜 can be defined. It is the [[Analysis/Sets|subset]] of the [[Analysis/PowerSet|power set]] of the sample space (𝒜 ''⊆ P(Ω)'') that satisfies three conditions: | |
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=== Subsets === Subsets are usually named with calligraphic uppercase letters, but that's not exactly practical in typed notes. Capital letters will be used instead. A subset of ''Ω'' is expressed as ''A ⊆ Ω''. === Intersections and Unions === The '''intersection''' of two sets is notated as ''A ⋂ B''; the '''union''' of two sets is notated as ''A ⋃ B''. The intersection of all subsets ''A,,i,,'' can be expressed as: {{attachment:intersection.svg}} The union of all subsets ''A,,i,,'' can be expressed as: {{attachment:union.svg}} ---- == Properties == A pair of sets are '''disjoint''' if there is no intersection, which is expressed as ''A ⋂ B = ∅'' ---- == Maps == Maps are usually named with blackboard bold letters, but that's not exactly practical in typed notes. Bold capital letters will be used instead. A map translates a (sub)set into a real number. A parallel to the functional expression of probability, ''p(A)'', is '''''P''': A -> R''. |
* Both the empty set and the full sample space are in the σ algebra; ''Ø, Ω ∈'' 𝒜 * The sigma algebra is closed on complementation; ''A ∈'' 𝒜 ''-> A^C^ ∈'' 𝒜 * The sigma algebra is closed upon countable unions; ''⋃ A,,i,, ∈'' 𝒜 where members of 𝒜 be indexed as ''A,,i,,'' * This indirectly requires that the sample space itself be [[Analysis/Cardinality|countable]]; ''i ∈ N'' |
σ Algebra
A σ algebra (sigma algebra) is a measurable set.
Contents
Description
A σ algebra is usually notated using a calligraphic uppercase letters, like 𝒜.
For a given sample space Ω, a σ algebra 𝒜 can be defined. It is the subset of the power set of the sample space (𝒜 ⊆ P(Ω)) that satisfies three conditions:
Both the empty set and the full sample space are in the σ algebra; Ø, Ω ∈ 𝒜
The sigma algebra is closed on complementation; A ∈ 𝒜 -> AC ∈ 𝒜
The sigma algebra is closed upon countable unions; ⋃ Ai ∈ 𝒜 where members of 𝒜 be indexed as Ai
This indirectly requires that the sample space itself be countable; i ∈ N