Probability Space

Definition

A probability space is a special type of measure space in which the whole space has measure 1. That is, a probability space is a measure space (ω,F,P) where:

P(Ω)=1.

The way in which probabilities are quantified using a measure space is clear from the following definitions.

Definition

Given the probability space (Ω,F,P):

  1. The underlying set Ω is called the sample space.
  2. Elements of the sample space Ω are called outcomes.
  3. Subsets of the sample space which are in the σ-algebra F are called events.
  4. The measure is called a probability function.
Theorem

For any S∈F, P(S)∈[0,1].

Note that sometimes a definition of the probability space is given so that P has codomain [0,1]. Here we take this to be a theorem so that in the strictest sense a probability space is a special case of a measure space.

Proof

This is a natural consequence of the fact that the measure of a subset of the space is less than or equal to the measure of the whole space, and the measure of the whole space is 1.