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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?

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up vote 2 down vote accepted

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:

Banach Tarski Paradox:

For reasons why there are some sets so pathological its nonsensical to assign them a measure/probability.

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$ \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 ...

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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.

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