Let $f:\mathbb{R}^n\to\mathbb{R}$ be an analytic function, and consider the semianalytic set $E:=\{\vec x\in\mathbb{R}^n|f(\vec x)=0\}$. Is the set $E$ automatically Lebesgue measurable?
Note: My question is possibly related to Are subanalytic sets measurable? but I get lost in the definition of subanalytic, so it would really help me out if someone could walk me through if/why measurability works for this simpler class of sets. If this is related to a known (named) theorem, I'd be more than happy to just understand the connection to that result, no need to prove the whole theorem.