How to define the k-dimensional Lebesgue measure $\lambda^{k,n}$ on $R^n$ when $k<n$

Question

Rene Schilling says

Which can be stated more generally:

Let $B\in\mathfrak{B}(\mathbb{R}^k)$ and let $\Phi: \mathbb{R}^k\to\mathbb{R}^n$ (where $k\leq n$) be a $C^1$-map for which there exists $Q\subset \mathbb{R}^n$ with $\lambda^{k,n}(Q)=0$ such that $\Phi(x_1)=\Phi(x_2)\in \Phi(B)\setminus Q\ \Rightarrow\ x_1=x_2$. Then $$\lambda^{k,n}(\Phi(B))=\int_B\sqrt{\det((D\Phi(x))^T D\Phi(x))} \lambda^{k,k}(dx)$$ where $\lambda^{k,n}$ denotes the $k$-dimensional Lebesgue measure for subsets of $\mathbb{R}^n$.

The problem is I don't know how Schilling's definition of $\lambda^k$ should be extented to define $\lambda^{k,n}$ in order to imply the more general Jacobi theorem.

How should this be done and/or are there any alternative references that treats the more general problem?

Answer 1

I'd recommend Francesco Maggi's book:
Sets of Finite Perimeter and Geometric Variational Problems -
An Introduction to Geometric Measure Theory

which is amazingly well written in my opinion. What you wrote is basically Theorem 8.1 of it.