When, if ever, is a probability space $(\Omega, \mathcal{F}, P)$ with $\mathcal{F} \neq 2^\Omega$ relevant? I understand the utility of working with $\mathcal{F}$ as the domain of $P$, however, all reasonable definitions of $P$ I can think of don't seem to care about other elements in $\mathcal{F}$ as long as its argument is a member.
Is there any situation that justifies the generality of writing $(\Omega,\mathcal{F},P)$? When would we want $\mathcal{F} \neq 2^\Omega$?
 A: One very important probability space is $\Omega=[0,1]$ with $\mathbb{P}$ being Lebesgue measure on $\Omega$.
If you accept the axiom of choice then not every subset of $[0,1]$ is Lebesgue measurable, so in this case $\mathcal{F}$ is either the $\sigma$-algebra of Lebesgue measurable subsets of $[0,1]$ or the smaller $\sigma$-algebra of Borel subsets of $[0,1]$.
A: Most of the time.
More precisely, there is absolutely no guarantee that a given measure on a $\sigma$-algebra can be extended to a measure on the field of all subsets of the underlying space, nor is there any natural "maximal" $\sigma$-algebra to which a given measure can be extended to.
Furthermore, even if it can be done, it hardly ever can be done in a way that preserves nice properties of the measure we had started with, like invariance under group action.
Moreover, these extensions (if they even exist) are usually highly non-constructive, unlike naturally occurring Borel measures, such as the Lebesgue measure, Haar measures and their relatives.
