Let $F:[0,1]\to[0,1]$ be a Lebesgue measure preserving function. Is $F$ almost surjective, i.e., the image of $F$ has interior measure one?

This question is motivated by the following observation. If $F$ satisfies the above and $X$ is uniformly distributed over $[0,1]$, then $F(X)\sim Unif[0,1]$, i.e., $F(X)$ seems to appear almost everywhere in [0,1]. I believe the answer is no, but a counterexample is nontrivial.

A relating post is https://math.stackexchange.com/questions/2612075/is-every-measure-preserving-function-almost-surjective but the counter example therein does not apply to my question. (The function constructed there has different domain and range.)