Suppose $n>1$, and let $U \subseteq \mathbb{R}^n$ be an open connected set.
Let $f$ be a real-analytic function on $U$. Suppose that $f=0$ on a subset of $U$ of positive measure. Is it true that $f$ vanishes identically?
I know that if $f=0$ on an open subset of $U$, then it is identically zero. (This is the identity theorem for real-analytic functions).
