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.

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 $$\Sigma$$.

When $$\left({\Omega, \Sigma, \Pr}\right)$$ is a discrete probability space, the probability measure is called the probability mass function.

When $$\left({\Omega, \Sigma, \Pr}\right)$$ is a continuous probability space, the probability measure is called the probability density function.