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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.

gt6989b
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2 Answers2

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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.

gt6989b
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  • 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"? –  May 08 '17 at 15:08
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    @user419308 yes, exactly. You can impose structure on a space, but don't think it is explicitly implied by using the word space. – gt6989b May 08 '17 at 15:10
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Measure Space. I enjoy that question of space. I think it might be an extension of an -tuple of objects involving some set, its powerset, and restrictions, also viewed as equipment or augmentation (at the mental level one is "adding" restrictions).

A probability space will be a particular case of a measure space, where the measure is a probability measure. The domain can come from many parts of mathematics, including Euclidian vector spaces.

The power set being the set of all subsets of the domain of the tuple.

Such a vocable would include any algebraic structure, I guess.. if being about a set, some operators on that set and some properties of that set and such operators. I wonder why people would not call a "group", also a space. Perhaps, some restrictions on the type of restrictions. or just history. I am not coming from abstract algebra. And speaking from long never revisited memory. I also lack symbolic math. touch typing skills.

I do agree that the invariance by translation from our intuitive Euclidian spaces, is something I would want for a space to have. I have had also difficulties, intuitively accepting finite unordered sets and their operations, to be called spaces. It may depend on the ordered nature of the "domain" of such tuple.

It is at least a natural human intuition thing to invoke when talking about any object (it has to be somewhere). I do not think it is actually a definable part of the mathematical theory (language body?)where they are used. And maybe, one could try to discern why op and I have the same discomfort calling things (structure, tuples) of finite nature for the domain, with no ties to "geometric" things, are not space. What about Euclidian spaces do we want to be also present in other "spaces"? I think ordered, and some local invariance, but would it have to be a continuum?

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