Definition:Probability Space

Definition

A probability space is a measure space $\left({\Omega, \Sigma, \Pr}\right)$ in which $\Pr \left({\Omega}\right) = 1$.

A probability space is used to define the parameters determining the outcome of an experiment $\mathcal E$.

In this context, the elements of a probability space are generally referred to as follows:

• $\Omega$ is called the sample space of $\mathcal E$;

• $\Sigma$ is called the event space of $\mathcal E$;

• $\Pr$ is called the probability measure on $\mathcal E$.

Discrete Probability Space

If $\Omega$ is a discrete sample space, then $\left({\Omega, \Sigma, \Pr}\right)$ is known as a discrete probability space.

Continuous Probability Space

If $\Omega$ is a continuum, then $\left({\Omega, \Sigma, \Pr}\right)$ is known as a continuous probability space.

Probability Function

The probability measure $\Pr$ on a probability space $\left({\Omega, \Sigma, \Pr}\right)$ can be considered as a function on elements of $\Omega$ and $\Sigma$.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 1.1, \ \S 1.4$