All of us know that "$X\sim U(a,b)$" means that both of $F_X$ and $f_X$ have a particular formulation.
But, since $X$ is a random variable, then there is a probabilistic space $(\Omega,\mathcal{F},P)$ such that $X:\Omega\to\mathbb{R}$ is a Borel measurable function. How can we obtain "the explicit rule" for this function $X$? That is: is there an explicitely expression for the map $\Omega\ni w\mapsto X(w)\in\mathbb{R}$?
What about others random variable?
Well, a very simple (even trivial) example would be the identity, i.e. $$(\Omega,~\mathcal{F},~\mathbb{P}) = \left((a,b),~\mathcal{B}_{\Omega},~\mathcal{U}_{\Omega}\right),~~ X = \mathrm{id}_{\Omega}.$$ However, a common idea is that the point of dealing with the distribution of a random variable $X$ is that we don't have to care about the underlying space $(\Omega,~\mathcal{F},~\mathbb{P})$ and the mapping $X$: Say, we take $(a,b) = (-1,1)$, then another example would be given by $$\Omega = (-2,-1)\cup[0,1],~\mathcal{F} = \mathcal{B}_{\Omega},~\mathbb{P} = \mathcal{U}_{\Omega}$$ with $$X:\Omega\rightarrow(-1,1),~\omega\mapsto\left\{\begin{array}{cl}\omega+1,&\omega<0,\\\omega,&\omega\geq 0.\end{array}\right.$$ Of course, this doesn't change anything about the behaviour of $X$ as a uniformly distributed r.v. and we could come up with endless (less simple) examples.
I suppose a related question that's more interesting is how you obtain uniformly distributed (or other distributions) r.v.s through transformation or combination of other r.v.s: For instance, there is a classical result saying that for statistical tests (under certain general conditions) the $p$-value will be uniformly distributed on $(0,1)$ under the null hypothesis. Or if $X,Y$ are iid geometrically distributed and you condtion on $\{X + Y = z\}$, then $X$ and $Y$ will be uniformly distributed on $\{0,1,\ldots,z\}$. Or also the Poisson distribution as a limit (in a sense) of binomial distributions...
If it is said that $X$ is a random variable and that $F$ denotes its CDF then this indeed means that some probability space $(\Omega,\mathcal A,P)$ exists and that $X$ is a Borel measurable function $\Omega\to\mathbb R$.
"What probability space then?" You might wonder. But in practice this is mostly not relevant. Also see this question (and its answer) to get a better view on what I mean.
If you kind of insist on explicitly mentioning such probability space then there are several possibilities to construct one.
By far the easiest is taking $(\Omega,\mathcal A,P)=(\mathbb R,\mathcal B(\mathbb R),P)$ where $B(\mathbb R)$ denotes the collection of Borel subsets of $\mathbb R$ , where $P$ denotes the probability measure that is induced by $F$ and where $X:\Omega=\mathbb R\to\mathbb R$ is the identity function.
