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| = σ Algebra Notation = | = σ Algebra = |
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| A '''σ algebra''' (sigma algebra) is a measurable set. | |
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== Sets and Subsets == The maximal set, which in probability applications is the '''sample space''', is notated as ''Ω''. The sample space could be a discrete set, like ''Ω = {heads, tails}''. It could be a set of discrete numbers, like ''Ω = '''N''''' (all real numbers). It could be a continuous range, like ''Ω = [0,1]''. === 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''. === Power sets === The power set of a set (''P(Ω)'') is the set of all subsets, including the empty set (''∅'') and the set itself (''Ω''). This becomes analagous to a probability function in descrete cases. === 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}} A pair of sets are '''disjoint''' if there is no intersection, which is expressed as ''A ⋂ B = ∅'' === Complements === The '''complement''' of a subset ''A'' is notated as ''A^c^''. |
<<TableOfContents>> |
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| == Sigma Algebras == | == Description == |
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| A '''σ algebra''' is usually named with calligraphic uppercase letters, but that's not exactly practical in typed notes. Capital letters will be used instead. | A σ algebra is usually notated using a calligraphic uppercase letters, like 𝒜. |
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| A σ algebra is notated as ''A ⊆ P(Ω)''. In other words, ''A'' is a subset of the power set of ''Ω''. | 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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| To qualify as a σ algebra, ''A'' also needs to satisfy three properties: * ''Ω'' is in ''A'' * ''A'' is closed upon complementation. For any subset, the complement of that subset is also in ''A''. * ''A'' is closed upon countable unions. ---- == 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: '''''M''': A -> '''R'''''. === Probability Measures === '''Probability measures''' are the primary use of maps with σ algebras. A parallel to the functional expression of probability, ''p(A)'', is '''''P''': A -> [0,1]''. |
* 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