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].