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 $\Sigma$.