Probability Space

A probability space is a measure space equipped with probability measure.

Description

A probability space is a triple (Ω, ℱ, P), composed of:

• a sample space Ω

• an event space ℱ, noting this is a σ algebra

• a probability measure P : ℱ -> [0,1] satisfying three conditions called the Kolmogorov axioms

A closely related concept is a random variable, which itself is just a map between a sample space and some measurable space (Ω, 𝒜): X : Ω -> 𝒜. When the sample space Ω is common to both the random variable and a probability space, it is possible to lift the probability measure P from the latter into the former. This pushforward measure is a probability distribution. The probability that the random variable X takes on the value x ∈ 𝒜 is given by P(X = x) = P({ω ∈ Ω | X(ω) = x}). The probability that the random variable takes on any value in the subset S ⊆ 𝒜 is given by P(X ∈ S) = P({ω ∈ Ω | X(ω) ∈ S}).

(It may be more appropriate to say that the random variable maps to a topological space. Practically speaking, probability is applied in two ways: the sample space is countable and discrete, or the sample space is uncountable but the random variable maps to real numbers. In either case, the Borel set of Ω is the smallest possible σ algebra that can be defined on Ω.)

In the case of a discrete random variable X, f is called a probability mass function and the final condition above is expressed as:

In the case of a continuous random variable X, f is called a probability density function and the final condition above is expressed as:

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