Editing Probability space (section)

=== Defining the events in terms of the sample space ===
If Ω is [[countable]], we almost always define <math> \mathcal{F}</math> as the [[power set]] of Ω, i.e. <math> \mathcal{F} = 2^\Omega</math> which is trivially a σ-algebra and the biggest one we can create using Ω. We can therefore omit <math> \mathcal{F}</math> and just write (Ω,P) to define the probability space.

On the other hand, if Ω is [[uncountable]] and we use <math> \mathcal{F} = 2^\Omega</math> we get into trouble defining our probability measure ''P'' because <math> \mathcal{F}</math> is too "large", i.e. there will often be sets to which it will be impossible to assign a unique measure. In this case, we have to use a smaller σ-algebra <math> \mathcal{F}</math>, for example the [[Borel algebra]] of  Ω, which is the smallest σ-algebra that makes all open sets measurable.