For a Lipschitz map $f: \mathbb{R}^n \to \mathbb{R}^m$ with $n \le m$, and for any measurable set $A \subset \mathbb{R}^n$, $$ \int_A J_n f , d\mathcal{L}^n = \int_{\mathbb{R}^m} \mathcal{H}^0(A \cap f^{-1}{y}) , d\mathcal{H}^n(y). $$
Federer writes in an extraordinarily precise, almost formalist style. Lemma 1.2 might reference a result from Appendix 2.3.1, which uses notation defined in Chapter 0, §4. You will flip pages (or scroll frantically) constantly. This is not a beach read. Why Bother? (The Allure of GMT) Geometric Measure Theory (GMT) was invented to solve one infuriating problem: How do you take the "surface area" of something that isn't smooth? federer geometric measure theory pdf
Last month, I finally decided to stop treating the PDF on my hard drive as a sacred artifact and actually opened it. Here is the view from the trenches. First, a note on the PDF. The original Springer “Grundlehren” edition runs 676 pages. The typesetting is pure late-60s elegance: no LaTeX, yet strangely beautiful. The PDFs floating around (legally purchased, of course) are usually clean scans, but they preserve the original’s dense theorems and famously terse proofs. For a Lipschitz map $f: \mathbb{R}^n \to \mathbb{R}^m$