Why is the sample space needed to define a probability space? To define a probability space we need de sample space, a collection from sample space that has to be a sigma-algebra, and a probability function.
But in order to the sigma-algebra being defined, It has to contain the sample space. So why can´t define a probability space just with de collection and the probability funcion?
 A: It is actually possible to start with the $\sigma$-algebra. This is not typical but it can be done by writing down a definition of an abstract $\sigma$-algebra (one that doesn't come equipped with an embedding into a $\sigma$-algebra of sets), exactly the way we define abstract Boolean algebras, abstract groups, abstract rings, etc. We get the usual advantage of abstraction, namely the ability to define quotients: for example it now makes sense to talk about the quotient of the Borel $\sigma$-algebra of $[0, 1]$ by the ideal of sets of Lebesgue measure zero. You can see some more details in this blog post by Terence Tao. I think there's another post where he uses abstract $\sigma$-algebras more heavily to prove some result in ergodic theory but I can't find it at the moment.
Doing the analogous thing in topology produces what is called the theory of locales which is somewhat better behaved in various ways.
In practice this doesn't seem to buy you enough that people care to make the switch, and it's convenient to use sample spaces to define $\sigma$-algebras, so that's what people do. But I think it's worth knowing that an alternative exists.
