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?
Best Answer
This is an example where $\mathcal{F} \neq \mathcal{P}(\Omega)$:
Let $\Omega = \{0, 1\}^{\Bbb{N}}$ be the set of sequences with values only $0$ and $1$. Also let $\mathcal{F} = \{\varnothing, E_0, E_1, \Omega \}$, where
$$E_{i} = \{ \omega \in \Omega : \omega(1) = i \}. $$
Then $(\Omega, \mathcal{F})$ is a measurable space. Moreover, if we define $\Bbb{P}(E_0) = \Bbb{P}(E_1) = \frac{1}{2}$ together with the additivity, then $\Bbb{P}$ becomes a probability measure on $(\Omega, \mathcal{F})$.
Here is an interpretation: Suppose we toss a fair coin infinitely. Then $\Omega$ denotes all the possible outcomes, where $0$ corresponds to 'head' and $1$ corresponds to 'tail'. If we only know the result of the first coin toss, then all the information at hand can be represented as the $\sigma$-algebra $\mathcal{F}$.
Though it may first seem an artifact, indeed this formulation arises quite often in the probability theory, especially when we want to change the set of information itself.