Reference request: Federer-Besicovitch structure theorem

Question

$\newcommand{\R}{\mathbf{R}}\newcommand{\H}{\mathcal{H}}$Federer-Besicovitch prove the following result.

Theorem: Let $E \subset \R^{N}$ be a purely $k$-unrectifiable set such that $E = \cup_{j=1}^\infty E_j$ and $\H^k(E_j)< +\infty$. Then $\H^k(\pi_{K}E) = 0$ for $\sigma$-almost every $k$-plane $K$ in $\R^N$, where $\sigma$ denotes the uniform measure on $O(N, k)$.

Brian White proves the theorem by induction, assuming the case $k = 1, N = 2$. I am wondering if there is a place where I can read a proof of the theorem for this base case? I cannot find a self contained proof (assuming general familiarity with Hausdorff measure and rectifiability) anywhere.

Falconer provides an additional source (see comments below), but the language employed is slightly different. Wondering if there are any more recent expositions available.

Thanks!

Answer 1

Edit: this answer doesn't answer the updated question but I'll leave it here for posterity.

The second page of Brian White's paper cites The Geometry of Fractal Sets by K. Falconer for a proof of Besicovitch's theorem (the $k=1$, $N=2$ case).

Answer 2

See Chapter 18, Theorem 18.1, Part 2 in

Mattila, Pertti, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability., Cambridge Studies in Advanced Mathematics. 44. Cambridge: Cambridge University Press. xii, 343 p. (1999). ZBL0911.28005.

The proof of the theorem occupies almost the entire chapter.