(Probability Space) Shouldn't $\mathcal{F}$ always equal the power set of $\Omega$? This is from the wikipedia article about Probability Space:

A probability space consists of three parts:

1- A sample space, $\Omega$, which is the set of all possible outcomes.
2- A set of events $\mathcal{F}$, where each event is a set containing zero or more outcomes.
3- The assignment of probabilities to the events, that is, a function $P$ from events to probability levels.


I can not think of a case where $\mathcal{F}$ is not equal to the power set of $\Omega$. What is the purpose of the second part in this definition then?

If $\Omega$ is an infinite set, you can run into problems if you try to define a measure for every set. This is a common issue in measure-theory, and the reason why the notion of a $\sigma-$algebra exists. See, for instance:
The Vitali Set: http://en.wikipedia.org/wiki/Vitali_set
Banach Tarski Paradox: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
For reasons why there are some sets so pathological its nonsensical to assign them a measure/probability.
