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σ algebra uses and re-uses many common statistics [[Statistics/ProbabilityNotation|notations]]. <<TableOfContents>> ---- |
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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]''. |
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=== 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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| 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^''. |
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| == Properties == | == Sigma Algebras == |
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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. | |
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| A pair of sets are '''disjoint''' if there is no intersection, which is expressed as ''A ⋂ B = ∅'' | 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. |
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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
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 Ai can be expressed as:
The union of all subsets Ai can be expressed as:
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 Ac.
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].