Let $f \in W^{2,p}(U)$ for $1 \le p \lt \infty$ where $U$ is a compact subset of $\mathbb R^2$ and $g \in L^{\infty}(U)$ with $g(f)=f$ almost everywhere on $U$. I have trouble understanding why the following holds:
Assume that the set $\{f=0\}$ is non empty and moreover that $g \lt 1$ in $\{f=0\}$. Then it has a strictly positive lebesgue measure.
I know that $\{f=0\}$ is a (relatively) closed set but that is not enough for the strict inequality. What do I miss? Is it because it contains an open set?
Any help/hints are much appreciated! Thanks in advance!