Continuous Measure Theoretic Probability concrete example I would like to translate some concrete examples of random variables into a measure theoretic environment but I have trouble understanding the material for continuous cases.
An example is where the random variable $X$ denotes the IQ of people which is clearly seen to be a continuous random variable.
Hence $X$ must be some function from a probability space
$$
X : (\Omega, \mathcal{F},P) \to \mathbf{R}
$$
where the real line $\mathbf{R}$ denotes the IQ value, $\Omega$ the outcome space, $\mathcal{F}$ the set of events and $P$ the probability of these events.
How should I be thinking of the outcome space $\Omega$?
Is $\Omega$ the set consisting of all people labeled with an IQ value?
Since $\Omega$ is the outcome space, I would say that $\Omega = \mathbf{R}$ as well (assuming that negative IQ exists) since these are the possible outcomes?
Is $X$ then simply the identity function? 
Other example: $X$ is the random variable indicating time of arrival of an airplane.
In this case, is $\Omega$ the set $\mathbf{R}$ where each $\omega \in \Omega$ represents a time a plane can arrive?
More generally, can we consider all random variables $X$ to be the identity map from $\mathbf{R}$ to $\mathbf{R}$ where the distribution of $X$ depends on $\mathcal{F}$ and $P$?
 A: In many applications an underlying probability space $(\Omega,\mathcal{F},\mathsf{P})$ is rather an abstraction. Typically, one just specifies the distribution function of a random variable, e.g. $X\sim N(0,1)$. Such a statement makes sense because there is always a probability space corresponding to a distribution function $F$ on $\mathbf{R}$. As you noticed, one may take $(\Omega,\mathcal{F})=(\mathbf{R},\mathcal{B}(\mathbf{R}))$, $X(\omega)=\omega$, and $\mathsf{P}$ s.t.
$$
\mathsf{P}(\{\omega:X(\omega)\le x\})=F(x).
$$
A: Often, in probability, we do not care too much about the underlying measure space --- the only properties of the random variables that matter are their distributions (including joint distributions).
(Note, this is quite different from the perspective of measure theory --- if you study measures themselves, rather than real random variables, then the properties of the measure space can matter a great deal.)
This means that, as noted by d.k.o., technically, it should be perfectly fine to do just what you say. However, with real-world examples, like the ones you have cited, there is a more natural choice of the outcome space.
In the IQ example, $\Omega$ would be either the set of all people (no labeling involved), or maybe the set of people alive at a given time.
In the plane example, $\Omega$ would be the set of all plane landings (maybe ever, maybe in a given time/space frame).
In both cases, $\Omega$ is finite, so it can certainly be represented as a set of real numbers, but it hardly represents the reality of things.
There is a caveat: unfortunately, this is more or less useless for applications. In practice, if you want to predict e.g. whether a given plane will be late (and by how much), this is not the space you want to look at, since you do not know in advance what time will the given landing occur. Instead, you use known data about past landings and conditions in which they occurred in order to build a model which is supposed to predict the landing time based on known factors (such as current weather patterns, current position of the plane etc.).
The only remotely concrete $\Omega$ you can use for this purpose is exactly what it says on the tin: the set of all possible outcomes! This does not, in general, afford a physical description. Depending on your view of physical reality, it may be finite, but even then, it is large enough that it is easier to treat it as an abstract (infinite) measure space. For most purposes, you can identify this space with the real numbers, but that is more obfuscating than enlightening, so there is not much of a point. Thus we go back to an abstract space. It doesn't matter what $\Omega$ is, except that it is a measure space which supports random variables with such and such distribution.
