= 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.

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== 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.

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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: '''''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]''.



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