Probability space

Definition

Given a measure space [ilmath](X,\mathcal{A},\mu)[/ilmath] where [math]\mu[/math] is a Probability measure, we recall that [math]\mu(X)=1[/math] (and we will write [ilmath]\mu[/ilmath] as [ilmath]\mathbb{P} [/ilmath] in line with notation).

Now [ilmath](X,\mathcal{A},\mathbb{P})[/ilmath] is a Probability space

A Probability space is usually denoted [ilmath](\Omega,\mathcal{A},\mathbb{P})[/ilmath], here:

Name	Symbol	Type	Description
State space Sample space	[ilmath]\Omega[/ilmath]	Set	All the different states one can have or samples one can take
Event space	[ilmath]\mathcal{A} [/ilmath]	[ilmath]\sigma[/ilmath]-algebra	The events we can have
Probability measure	[ilmath]\mathbb{P} [/ilmath]	Function [math]\mathbb{P}:\mathcal{A}\rightarrow[0,1]\subset\mathbb{R}[/math]	Assigns probabilities to events

Example

Discrete probability space

Let us consider two die being thrown as our state or sample space - I prefer state because it is the set of states the experiment may take.

Then:

Part of prob. space	Definition	Comment
State space	[math]\begin{array}{lr} \Omega=\{ & (1,1), & (1,2), & \cdots &, (1,6),\\ & (2,1), & (2,2), & \cdots &, (2,6), \\ & \vdots \\ & (6,1), & (6,2), & \cdots &, (6,6) & \} \end{array}[/math]	The set of all possible states, there are 36 all-together.
Event space	[math]\mathcal{P}(\Omega)[/math] (see power set) = all subsets of [ilmath]\Omega[/ilmath]	Union works as or, for example [math]\{(1,2),(3,4)\}[/math] is the event that we get [ilmath](1,2)[/ilmath] or [ilmath](3,4)[/ilmath]
Probability measure	[math]\mathbb{P}:\mathcal{P}(\Omega)\rightarrow[0,1]\subset\mathbb{R}[/math] where: [math]\mathbb{P}(A)\mapsto \frac{1}{36}|A|[/math]	Clearly [math]\mathbb{P}(\Omega)=1[/math] and this is a measure!

This example alone isn't very interesting, it becomes interesting when one considers the random variable which could be for example the sum of the values shown on the die. That example is used on the random variable page.

Continuous probability space

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TODO: Think of example - Normal?

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See also

• Measure
• Measure Theory