(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?

$ \mathcal{F} $ is only the events we'd like to consider, not just all events.

The σ-algebra $ \mathcal{F} $ is a collection of all and only events (not necessarily elementary) we would like to consider.

On the event page it notes that

... all elements of the power set of the sample space are defined as events ...

