It recently came to my knowledge that, even though there is no way to uniformly choose a random real number (since the Lebesgue measure is not a probability measure), there is a canonical way to choose a random sequence of points in $\mathbb{R}^n$ such that the distribution of each point is in some sense uniform (a so-called homogeneous Poisson point process).
I'm not a probabilist, so this looks like dark wizardry to me. Is there a conceptual reason for which "choose a random point in $\mathbb{R}^n$ uniformly" has no chance of making any sense, but "choose a random sequence in $\mathbb{R}^n$ uniformly" has a pretty reasonable meaning ?
Let's take the simple example of a homogeneous Poisson process on $[0,\infty)$ with intensity $\lambda$. Think of $[0,\infty)$ as time and the realization of the Poisson process is the sequence of times where a given event occurs: $x_1,x_2,\dots$ Let $X_1,X_2,\dots$ the corresponding random variables. So $X_1$ is the time of the first occurence of the event, and so on.
The fact that $N$ is a Poisson process implies that the number of events that occur after time $a$ and before time $b$ is Poisson distributed, with parameter $\lambda (b-a)$. We have $$N((a,b])\sim\mathcal{P}(\lambda(b-a))$$
Here comes uniformity: given that $N((a,b])=n$ (i.e. there are $n$ occurences of the event in the time interval $(a,b]$), the times of occurence between $a$ and $b$ are uniformly distributed over $(a,b]$. To be more precise, for some $k$, we have $a<X_{k+1}\lt X_{k+2}\lt\dots\lt X_{k+n}\leq b$.
As this is valid for any interval $(a,b]$, we can say that in some sense, a relaization of $N$ is a random sequence of $[0,\infty)$ chosen uniformly.
Another way to explain it:
We denote $N([0,t])$ by $N_t$. Given that $N_t=n$, if you want to simulate the $n$ first times of arrival $X_1,\dots,X_n$, you can just simulate $U_1,\dots,U_n$ uniformly on $[0,1]$, re-order them, and multiply them by $t$.