In most probability theory texts you'll see a random variable definition that goes something like this:
A probability space is a triple $(\Omega, \mathcal{F}, P)$, a real valued random variable is a function $X:\Omega \rightarrow \mathbb{R}$
The computer scientist in my head can't get around what seems to be a type-error. Is a random variable dependent on $P$ or not. My intuition says it is, which is why we say things like 'uniformly distributed random variable', but in this case shouldn't a random variable be a function from the probability space, i.e. $X:(\Omega, \mathcal{F}, P) \rightarrow \mathbb{R}$. In fact this is how it is defined in Wolframs mathworld.
Alternatively, one could imagine in a functional programming language a function which counts the number of heads from a set of coin tosses. This function could be independent of whether or not the coin is fair or biased, etc, and hence in some way independent of $P$.
I understand some of this may be just for convenience when speaking, but this is how a random variable is rigorously defined, so I believe I must be missing something. In other words, my question is, is a random variable associated with probabilities or independent of it.
Best Answer
Your observation is reasonable, but your suggested cure for the problem, making $(\Omega,\mathcal F,P)$ the domain of $X$, won't work because the domain of a function needs to be a set (or a type or something like that). My impression is that, when people refer to a function $X:\Omega\to\mathbb R$ as a random variable, they always do so in the context of a probability-space structure ($\mathcal F$ and $P$) on $\Omega$. If no such structure is given, then I wouldn't call $X$ a random variable. And if there is uncertainty about which structure is intended, then, as you said, notions like "distribution" of $X$ will not be well-defined.
If I had to formalize the notion of random variable, in Bourbaki style, I would probably say that a random variable is a pair consisting of a probability space $(\Omega,\mathcal F,P)$ together with a function $X:\Omega\to\mathbb R$. As with many mathematical concepts, one often omits mentioning part of an entity (in this case the probability space) when it is understood from the context.