# Definition:Probability Space

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## Definition

A **probability space** is a measure space $\struct {\Omega, \Sigma, \Pr}$ in which $\map \Pr \Omega = 1$.

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

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

- $\Omega$ is called the sample space of $\EE$

- $\Sigma$ is called the event space of $\EE$

- $\Pr$ is called the probability measure on $\EE$.

Thus it is a measurable space $\struct {\Omega, \Sigma}$ with a probability measure $\Pr$.

### Discrete Probability Space

Let $\Omega$ be a discrete sample space.

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

### Continuous Probability Space

Let $\Omega$ be a continuum.

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

## Also see

- Results about
**probability theory**can be found here.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.1$: Experiments with chance - 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.4$: Probability spaces - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $4.2$