Probability Space - Definition

Definition

In short, a probability space is a measure space such that the measure of the whole space is equal to one.

The expanded definition is following: a probability space is a triple consisting of:

• the sample space Ω — an arbitrary non-empty set,
• the σ-algebra ⊆ 2Ω (also called σ-field) — a set of subsets of Ω, called events, such that:
  • contains the empty set: ,
  • is closed under complements: if A∈, then also (Ω∖A)∈,
  • is closed under countable unions: if A_i∈ for i=1,2,..., then also (∪_iA_i)∈
    • The corollary from the previous two properties and De Morgan’s law is that is also closed under countable intersections: if A_i∈ for i=1,2,..., then also (∩_iA_i)∈
• the probability measure P: → — a function on such that:
  • P is countably additive: if {A_i}⊆ is a countable collection of pairwise disjoint sets, then P(⊔A_i) = ∑P(A_i), where “⊔” denotes the disjoint union,
  • the measure of entire sample space is equal to one: P(Ω) = 1.