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 Ai∈ for i=1,2,..., then also (∪iAi)∈
- The corollary from the previous two properties and De Morgan’s law is that is also closed under countable intersections: if Ai∈ for i=1,2,..., then also (∩iAi)∈
- the probability measure P: → — a function on such that:
- P is countably additive: if {Ai}⊆ is a countable collection of pairwise disjoint sets, then P(⊔Ai) = ∑P(Ai), where “⊔” denotes the disjoint union,
- the measure of entire sample space is equal to one: P(Ω) = 1.
Read more about this topic: Probability Space
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