I am aware of the statement that a open set in $\mathbb{R}^n$, if it is star-like, is diffeomorphic to $\mathbb{R}^n$, although this is apparently not so easy to prove. I am wondering if a weaker statement exists. Namely, if $U$ is an open set of $\mathbb{R}^n$, does there always exist a diffeomorphism $\phi : U\to\mathbb{R}^n\setminus N$ where $N$ is a Lebesgue-negligible set?
Edit: Here is a reference the claim that star-shaped open sets are diffeomorphic to $\mathbb{R}^n$.