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Given that a "sample space" is defined as the set of possible outcomes of a given random experiment, is there a fundamental reason to use the term "sample space" instead of "sample set" in probability and statistics or is it just from historical or convenience reasons? In other words, do sample spaces have properties that distinguish them from being just sets, and if not, what is the (historical) origin of using "sample space" instead of "sample set"?

The only explanation that I have found so far is from the book "Cambridge 2 Unit Mathematics Year 12 Enhanced Version" by William Pender, David Saddler, Julia Shea and Derek Ward here:

The reason for using the word 'sample space' rather than 'sample set' is that the sample space of a multi-stage experiment takes on some of the characteristics of a space. In particular, the sample space of a two-stage experiment can be displayed on a two-dimensional graph, and the sample space of a three-stage experiment can be displayed on a three-dimensional graph.

However, I find the explanation rather lightweight/anecdotal.

The Quark
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    Probability and Statistics was more seen like Mechanics and Physics in the 19th and early 20th century (Kolmogorov's axioms didn't come in until 1930s!). It could be that "sample space" was terminology akin to "state space" (of classical mechanics), of course, this is just a guess. At any rate, it does not matter how you call it (in Markov chains you do say "state space" because those were thought as the probabilistic analog of classical mechanics "only the current state matters to infer about the future"). – William M. Apr 25 '22 at 17:17

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While sample set could also be worthwhile terminology, it is worth recalling that any space in mathematics is always a set with some sort of operations defined.

For example, in the theory of stochastic processes, we could let $\Omega = \mathcal{D}(\mathbb{R}_+, \mathbb{R})$, where $\mathcal{D}(\mathbb{R}_+, \mathbb{R})$ is the Skorokhod space of real-valued cádlág processes on the positive real line. In this sense, a stochastic process $X: \Omega \rightarrow \mathbb{R}$ can in be understood to be a map that picks some arbitrary path and gives the real value of that path at various time points. The Skorokhod space is a vector space, so in this sense it makes sense to consider $\Omega$ as a space, not just a set.

Since we often deal with such situations in probability, it could be surmised that people generally have accepted the terminology since $\Omega$ often is, in fact, a space.

Forgottenscience
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  • First of all, thanks for your answer and very quick reply. As a comment, I understand that some sets, complemented with some appropriate structure and operations, are also spaces. But this might not generally be the case. So in a sense, "sample set" is more general than "sample space" and "sample space" might therefore be restricted to those particular cases where the set turns out to also be a space. So then, again, why is it the contrary? – The Quark Aug 30 '15 at 12:41
  • So it seems you mean an "algebraic structure" when you say "space", not a "geometric space", right? https://math.stackexchange.com/a/174297/340174 and https://math.stackexchange.com/a/177943/340174 – Make42 Jun 04 '20 at 21:01