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| = σ Algebra Notation = | = σ Algebra = A '''σ algebra''' (sigma algebra) is a set of measurable sets. |
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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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| Subsets are usually named with calligraphic uppercase letters, but that's not exactly practical in typed notes. Capital letters will be used instead. | 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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| A subset of ''Ω'' is expressed as ''A ⊆ Ω''. | * 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'' |
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| ---- | The smallest possible σ algebra is given by condition 1 literally: ''{Ø, Ω}''. The largest possible σ algebra is of course ''P(Ω)'' itself. All possible σ algebra are bounded by these two extrema. |
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| == Properties == | === Properties === |
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| The intersection of two sets is notated as ''A ⋂ B''; the union of two sets is notated as ''A ⋃ B''. | For a given sample space ''Ω'', and some number of σ algebras 𝒜'',,i,,'' defined on it, ''⋂,,i,,'' 𝒜 '',,i,,'' is also a σ algebra. |
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| A pair of sets are '''disjoint''' if there is no intersection, which is expressed as ''A ⋂ B = ∅'' | It follows that for any ℳ ''⊆ P(Ω)'', the smallest possible σ algebra that contains ℳ (or the σ algebra generated by ℳ) is given by the intersection of every 𝒜 larger than ℳ. This operation is notated as: |
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| ---- | {{attachment:gen.svg}} |
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| The '''Borel σ algebra''' defined on ''Ω'' is the smallest possible σ algebra generated by the open sets of ''Ω''. These open sets form the topology of ''Ω'', notated as ''τ''. It follows that: | |
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== 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. This can be expressed as '''''P''': Ω -> R''. |
{{attachment:borel.svg}} |
σ Algebra
A σ algebra (sigma algebra) is a set of measurable sets.
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
The smallest possible σ algebra is given by condition 1 literally: {Ø, Ω}. The largest possible σ algebra is of course P(Ω) itself. All possible σ algebra are bounded by these two extrema.
Properties
For a given sample space Ω, and some number of σ algebras 𝒜i defined on it, ⋂i 𝒜 i is also a σ algebra.
It follows that for any ℳ ⊆ P(Ω), the smallest possible σ algebra that contains ℳ (or the σ algebra generated by ℳ) is given by the intersection of every 𝒜 larger than ℳ. This operation is notated as:
The Borel σ algebra defined on Ω is the smallest possible σ algebra generated by the open sets of Ω. These open sets form the topology of Ω, notated as τ. It follows that:
