σ Algebra

A σ algebra (sigma algebra) is a set of measurable sets.

Description

A σ algebra is usually notated using a calligraphic uppercase letters, like 𝒜.

For a given sample space Ω, a σ algebra 𝒜 can be defined. It is the subset of the power set of the sample space (𝒜 ⊆ P(Ω)) that satisfies three conditions:

• Both the empty set and the full sample space are in the σ algebra; Ø, Ω ∈ 𝒜

• The sigma algebra is closed on complementation; A ∈ 𝒜 -> A^C ∈ 𝒜

• The sigma algebra is closed upon countable unions; ⋃ A_i ∈ 𝒜 where members of 𝒜 be indexed as A_i, i ∈ N

The smallest possible σ algebra is given by condition 1 literally: {Ø, Ω}. The largest possible σ algebra is of course P(Ω) itself. All possible σ algebra are bounded by these two extrema.

Properties

For a given sample space Ω, and some number of σ algebras 𝒜_i defined on it, ⋂_i 𝒜 _i is also a σ algebra.

It follows that for any ℳ ⊆ P(Ω), the smallest possible σ algebra that contains ℳ (or the σ algebra generated by ℳ) is given by the intersection of every 𝒜 larger than ℳ. This operation is notated as:

Relation to Borel Sets

The Borel set (or Borel σ algebra) defined on Ω is the smallest possible σ algebra generated by the topology of Ω. It follows that:

The Borel set is equal to the power set only when Ω is discrete and countable. Otherwise, the Borel set is always a subset of the power set.

CategoryRicottone