If $x$ is a one dimensional random variable defined over some segment $[a,b],$ and $F$ is an atom-less, continuous distribution over it, then I know that $Pr_F(x=\tilde x)=0$ for every $\tilde x\in[a,b]$ and that for every $\varepsilon>0,$ $Pr_F(\tilde x-\varepsilon<x<\tilde x+\varepsilon)>0.$ I wnat to extend this notion to higher dimensions.
For example, I want to randomly choose a tuple $(x,y)$ out of some interval ${(x,y)| x\in[0,H],y\in[0,H]}.$ I am drawing using an atom-less distribution $F_{xy}.$ That is $Pr_F((x,y)=(\tilde x,\tilde y))=0$ for every $\tilde x,\tilde y.$
What is the probability of drawing a pair $(x,y=g(x))$ where $g$ is a function of $x$?
In General, what is the probability of randomly drawing a point in some function in $\mathbb{R}^n$ out of a space in $\mathbb{R}^{n+1}$?
I suspect that the probability of this event should be zero, how can this be proved or contradicted?