Axioms of Probability

Ingredients of Probability Spaces

Definition 1

A collection ℱ of subsets of a set Ω is called a ring on Ω if it satisfies the following conditions:

• 1. A, B ∈ ℱ ⇒ A ∪B ∈ ℱ,

• 2. A, B ∈ ℱ ⇒ A ∖ B ∈ ℱ.

A ring ℱ is called an algebra if Ω ∈ ℱ.

Definition 2

A ring ℱ on Ω is called a σ-ring if it satisfies the following additional condition:

• 3. For every countable family ${({A}_{n})}_{n\in \mathbb{N}}$ of subsets of ℱ: ${\bigcup \nolimits }_{n\in \mathbb{N}}{A}_{n} \in \mathcal{F}$.

A σ-ring ℱ on Ω is called a σ-algebra (or σ-field) if Ω ∈ ℱ.

Proposition 1

The following properties hold:

• 1. If ℱ is a σ-algebra of subsets of a set Ω, then it is an algebra.

• 2. If ℱ is a σ-algebra of subsets of Ω, then

  • For any countable family ${({E}_{n})}_{n\in \mathbb{N}\setminus \{0\}}$ of elements of ℱ: ⋂ ^∞ _{n = 1} E _n ∈ ℱ,

  • For any finite family (E _i )_{1 ≤ i ≤ n} of elements of ℱ : ⋂ ⁿ _{i = 1} E _i ∈ ℱ,

  • B ∈ ℱ ⇒ Ω ∖ B ∈ ℱ.

Definition 3

Every pair (Ω, ℱ) consisting of a set Ω and a σ-ring ℱ of subsets of Ω is a mea...