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| <<TableOfContents>> | == 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. |
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| == Sets and Subsets == | == Sigma Algebras == |
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| The maximal set, which in probability applications is the '''sample space''', is notated as ''Ω''. | 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. |
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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''. | |
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| To qualify as a σ algebra, ''A'' also needs to satisfy three properties: | |
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| === Subsets === | * ''Ω'' is in ''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. | * ''A'' is closed upon complementation. For any subset, the complement of that subset is also in ''A''. |
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| 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 = ∅'' |
* ''A'' is closed upon countable unions. |
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| Maps are usually named with blackboard bold letters, but that's not exactly practical in typed notes. Bold capital letters will be used instead. | '''Maps''' are usually named with blackboard bold letters, but that's not exactly practical in typed notes. Bold capital letters will be used instead. |
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| A map translates a (sub)set into a real number. A parallel to the functional expression of probability, ''p(A)'', is '''''P''': A -> R''. | 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]''. |
σ Algebra Notation
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.
Sigma Algebras
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 notated as A ⊆ P(Ω). In other words, A is a subset of the power set of Ω.
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].