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 A_i can be expressed as:

The union of all subsets A_i 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 A^c.

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].

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