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Originally, the word 'space' reffered to the "boundless three-dimensional extent", as Wikipedia tells us. In modern mathematics, 'space' is used in a more general sense, referring to a set with some added structure, so that besides $\mathbb R^3$, we can also consider $\mathbb R^n$ for every other natural number $n$ to be a space, can consider the set of functions $A\to B$ between any two sets together with additional structure (such as addition $(f+g)(x)=f(x) + g(x)$) as a space, and so on.

In some sense, spaces are the same as structures, but in my understanding the difference between structures and spaces is that we consider the latter to be a special case of the former that have a 'geometric' nature (of course, 'geometric' here doesn't have a formal definition, and is just used informally).

Now, vector spaces, metric spaces, projective spaces, measurable spaces, ... clearly are geometric in its nature, and thus deserve to be called 'spaces'.

But why the hell do we, in probablity theory, call the set of all outcomes the 'sample space'? It's just defined to be the set of all possible outcomes/results of (random) experiment—without additional structure! I can sort of understand why we say 'probability space', I think it's because it's a special case of a measure space. But the sample space alone ... why is it called a 'space'? It hasn't additional structure and isn't geometric in nature.

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    $\begingroup$ Alliterations are awesome. $\endgroup$ – user307169 May 8 '17 at 15:03
  • $\begingroup$ @tilper: What do you mean, and what does it have to do with my question? $\endgroup$ – user401895 May 8 '17 at 15:06
  • $\begingroup$ In this cases, "space" is an "ambient" and thus it takes the place of domain. $\endgroup$ – Mauro ALLEGRANZA May 8 '17 at 15:07
  • $\begingroup$ Related post: what-does-a-space-mean. $\endgroup$ – Mauro ALLEGRANZA May 8 '17 at 15:10
  • $\begingroup$ @MauroALLEGRANZA: thanks. $\endgroup$ – user401895 May 8 '17 at 15:14
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Loosely, the mathematical notion of space is a cavity where the objects exist. In that sense, sample space is also a kind of space, in which the experiment's results can be described.

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  • $\begingroup$ So, do you think that besides "structure that has a geometric nature" the word 'structure' also just means "domain"/"cavity" and is thus used as a synonym of "set"? $\endgroup$ – user401895 May 8 '17 at 15:08
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    $\begingroup$ @user419308 yes, exactly. You can impose structure on a space, but don't think it is explicitly implied by using the word space. $\endgroup$ – gt6989b May 8 '17 at 15:10

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