I am currently reading this paper by Epstein. I need help with understanding the proof. Specifically, I have the following two questions.
- Let $w\colon G\to H$ be an analytic mapping between connected Lie groups $G$ and $H$. How do we prove that the preimage of a point is either the whole of $G$ or has Haar measure zero in $G$? The paper says one should take $G$ to be an open subset of $\mathbb{R}^m$ and use induction on $m$ and Fubini's theorem, but this doesn't really convince me (I cannot fill the details).
- The whole paper seems to assume that $G$ is analytic manifold, whereas the usual assumption is that Lie group is smooth manifold. Does this mean that the result of the paper hold only for real-analytic Lie groups, or this assumption is not actually a restriction and in some way we also have the claim for the classical real Lie groups?
Edit: Actually, regarding (1), I don't see why the usual $t\in\mathbb{R}\mapsto e^{-1/t^2}1_{t>0}\in\mathbb{R}$ is not a counterexample.
Try this:
Work in Euclidean space and write $w=(w^1(x_1,...x_n),...w^m(x_1,...,x_n))$. Write any $w^i$ as a power series in $x_n$ with coefficients as power series in the other variables: $$w^i=\sum_j c_j(x_1,...,x_{n-1})x_n^j$$
Then by Fubini, if $w^i$ vanishes on a set of positive measure, there exists a set of positive measure $A\subset \mathbb{R}^{n-1}$ such that for any $\overline{a} \in A$, the analytic function $\mathbb{R}\rightarrow \mathbb{R}, t \mapsto w^i(\overline{a}, t)$ vanishes on a set of positive measure. But then this $w^i(\overline{a}, \cdot )$ vanishes identically, so that each coefficient $c_j$ vanishes on $A$. Then by induction, each $c_j$ is identically zero on $\mathbb{R}^{n-1}$, so that $w^i$ is identically zero.