I'm trying to understand Serre's mass formula, but it's not very clear to me what does it mean to integrate over a subset of $L$, or $K^n$, and in particular why it holds an analogue of the usual change of variable formula.
The setting is the following: $K$ is a local field and $L$ a totally ramified extension; then we have a map $\varphi$ from the set $\Pi$ of uniformizers of $L$ to the set $P$ of Eisenstein polynomials over $K$ for which $K[x]/(f)\cong L$.
We give coordinates $(x_1,\dots,x_n)$ on $P$ using the basis $1,x,\dots,x^{n-1}$ and $(y_1,\dots, y_n)$ on $\Pi$ by choosing an integral basis.
Serre then says that $\varphi$ is an étale covering of degree $d$, so we have $\nu(P)=\frac1d\cdot|\text{Jac}\varphi|\cdot\mu_L(\Pi)$, where the Jacobian is computed wrt the chosen coordinates, and the measures $\nu=dx_1\cdots dx_n$ and $\mu_L=dy_1\dots dy_n$ are derived from the Haar measure on $K$.
How can I make sense of this "integrating over a manifold over a field"? Is there some book/article that explains some of this?