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 \((\omega, \mathcal{F}, P)\) where:

\[ P(\Omega) = 1.\]

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

Definition

Given the probability space \((\Omega, \mathcal{F}, \mathbb{P})\):

  1. The underlying set \(\Omega\) is called the sample space.
  2. Elements of the sample space \(\Omega\) are called outcomes.
  3. Subsets of the sample space which are in the \(\sigma\)-algebra \(\mathcal{F}\) are called events.
  4. The measure is called a probability function.
Theorem

For any \(S \in \mathcal{F}\), \(\mathbb{P}(S) \in [0, 1]\).

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 \(1\).