I would like to understand what Neukirch means when he writes down
$$ vol_{canonical}(X) = 2^s vol_{Lebesgue}(f(X)) $$
(Neukirch, Algebraic Number Theory. Pg 31)
I will write down the details at the end of the post.
I think it suffices to understand the following situation:
Let $V$ be a Euclidean vector space.
This means that $V$ is a finite-dimensional real vector space with symmetric, positive-definite bilinear form $\langle ~,~ \rangle : V \times V \to \mathbb{R}$. Let $\mu$ be the Haar measure of a Euclidean vector space (such that the cube spanned by the orthonormal basis has measure 1).
Let $e_i \in \mathbb{R}^n$ be the vector with $1$ on $i$th coordinate and $0$ everywhere else. Let $\Phi$ be the cube spanned by $e_i$. Then $\mu_{Leb}(\Phi)=1$ where $\mu_{Leb}$ is the Lebesgue measure on $\mathbb{R}^n$.
Question: Let $f: V \to \mathbb{R}^n$ be a linear isomorphism such that $$ \mu(f^{-1}(\Phi)) = c $$ or equivalently $$ \mu(f^{-1}(\Phi))= c \mu_{Leb}(\Phi) $$ Then for any measurable set $X \subset V$, $$ \mu(X) = c \mu_{Leb}(f(X)) $$
Details: Let $K/\mathbb{Q}$ be a number field with $[K:\mathbb{Q}]=n$. Let $K_\mathbb{R}$ be the set of $n$-tuple of $\mathbb{C}$ indexed by $Hom(\mathbb{Q}, \mathbb{C})$ with the following properties. If $\rho_1, \ldots, \rho_r$ are real embeddings of $K$ and $\tau_1, \overline{\tau_1}, \ldots, \tau_s, \overline{\tau_s}$ are the complex embeddings, then elements $(z_\sigma)$ in $K_\mathbb{R}$ is of the form $$ z_\rho \in \mathbb{R} \text{ and } z_\overline{\tau} = \overline{z_\tau} $$ where $\sigma$ varies over $Hom(K,\mathbb{C})$. $K_\mathbb{R}$ becomes a finite-dimensional real vector space, so it has a Haar measure $vol_{canonical}$. Let $f: K_\mathbb{R} \to \mathbb{R}^n$ be a linear isomorphism defined by $(z_\sigma) \mapsto (x_\sigma)$ where $$ x_\rho = z_\rho \text{ and } x_\tau = Re(z_\tau), x_\overline{\tau} = Im(z_\tau) $$ $K_\mathbb{R}$ has a symmetric, positive-definite bilinear form $$ \langle z, w \rangle = \sum_\sigma z_\sigma \overline{w_\sigma} $$ which transfers to $\mathbb{R}^n$ by $$ (x,y) = \sum_\sigma \alpha_\sigma x_\sigma y_\sigma $$ where $\alpha_\rho = 1$ for real $\rho$ and $\alpha_\tau = 2$ for complex $\sigma$.
Considering the same cube $\Phi$ as above, I was able to see that $vol_{canoncial}(f^{-1}(\Phi)) = 2^s = 2^s vol_{Lebesgue}(\Phi)$. But I am not sure how to pass this to general measurable set $X \subset V$.