In 1 it was proven that
Let $g,f: \mathbb{R}^n \to \mathbb{R}$ be two real-analytic functions. Suppose, that $g(x)=f(x)$ on a set $E$ of positive Lebesgue measure. Then, $f(x)=g(x)$ for all $x \in \mathbb{R}^n$.
My question is whether this result still holds for vector-valued (real) analytic functions $f: \mathbb{R}^m \rightarrow \mathbb{R}^n $?
My proof idea is as follows: We look at the component functions $f=(f_1,...,f_m)$. If every component function is analytic, then also $f$ is analytic. Hence if the identity theorem holds for each component, it also holds for the whole function $f$.
