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Let's say we have a probability space $(\Omega, \sigma, P)$. and a random variable $Y:\Omega\to \mathbb R$

$P$ is of course a mapping $P:\Omega\to \mathbb R$. Now, Omega is not necessarily a topology, but we can simply make the additional assumption that $\Omega$ also has a topology $T$ defined on it. This means that we can talk about whether $Y$ is a continuous function from $\Omega \to \mathbb R$.

The thing is, I usually think of the sample space $\Omega$ as something that "exists in the background", but that is not necessarily what we care about. We really care about the random variables, and when we define probability density functions on them for example, then the sample space is not in view at all (we only see the space of values that the random variable can take (usually $\mathbb R^n$), and the space that the probability density can take (which is $\mathbb R^+$).

So that's why I'm wondering, is there an intuitive interpretation of the random variable being a continuous map from $\Omega$ to $\mathbb R$?

Bonus: If the topology is a differentiable topological space, is there an interpretation of the random variable being a differentiable function?

EDIT: Obviously this will depend on the topology. My question implicitly asks whether there is a topology such that there is such a meaningful interpretation. That is, the interpretation doesn't have to be generally applicable to all topologies and all probability spaces in the same way.

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  • $\begingroup$ Not that I would be aware of. $\endgroup$ – Did Dec 17 '17 at 8:04
  • $\begingroup$ In light of your edit, I believe the intended question is not "assuming $\Omega$ has a topology, is there a good interpretation of continuity?", but rather, "are there any prominent examples of sample spaces $\Omega$ with natural choices of topology under which the notion of continuity has a useful probabilistic interpretation?". $\endgroup$ – Will R Dec 17 '17 at 10:29
  • $\begingroup$ IIRC, you can always tweak the sample space without changing the sigma algebra or measure so that $\sigma$ is actually a topology on $\Omega$. Some approaches to measure theory don't bother with a sample space at all; see "measurable locale". $\endgroup$ – Hurkyl Dec 17 '17 at 12:41
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I find it a good exercise from time to time to stop in any probabilistic or statistical example and try to imagine how a suitable $\Omega$ would look like or be defined. I usually end up thinking of different subsets of a given population of persons, or companies, whales, pins, apples, neutrons, stars... whatever. For such discrete sample spaces the analysis of this question would be kind of dull, having to fall perhaps in a discrete topology.

More interesting and complex are the cases where I feel the need to think of $\Omega$ as the set of possible states of the world at a given time, with all the detail in the description that one could imagine and where the "coordinates" tend to be continuous.

Given an "intuitive" topology for $\Omega$ (and then choosing as $\sigma$-algebra the corresponding Borel $\sigma$-algebra), the continuity of $X\colon \Omega \to \mathbb R$ would have the usual interpretation: that "small/infinitesimal" changes in the state of the world imply "small/infinitesimal" changes in the value of $X$.

Or better: given $\omega_0 \in \Omega$ you always can confine the value of $X$ to any neighborhood of $X(\omega_0)$ that you may choose (intuitively: as "small" as you want), by restricting the possible "states of the world" $\omega$ to a certain ("small") neighborhood of $\omega_0$. You can be sure that $X$ would had been very similar to $X(\omega_0)$ if the state of the world after the random experiment would have happened to be close enough to $\omega_0$.

In this context, suppose that the "state of the world" at the required level of detail can be described (at least ideally) by a (possibly astronomical) finite number $N$ of continuous variables (representing real-world magnitudes) probably each of them within certain bounds, an let it be endowed with the restriction of the usual topology of $\mathbb R^N$. In this scheme, if we define a "continuous r.v." $X$ (that is, continuous in the sense that $F_X$ is a continuous function), I guess that in order to have a meaningful "real-world" interpretation for $X$, the function $X(\omega)$ will have to be continuous (perhaps just a.e.?).

Nevertheless, this won't be the case for a discrete r.v., at least no for the aforementioned topologies. And regarding differentiability... I wouldn't know how to extend these ideas, if possible.

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