# Reference request: Federer-Besicovitch structure theorem

$$\newcommand{\R}{\mathbf{R}}\newcommand{\H}{\mathcal{H}}$$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!

## 1 Answer

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).

• Agreed, I've looked at the reference, and while it does seem to contain the proof, the language and terminology used is fairly different. Of course I can try translating the result into the modern language used in Simon's book (which is my main reference), but curious to see if there are any sources other than Falconer. – Drew Brady Nov 5 '18 at 18:31
• In that case I'm not sure, sorry. – Umberto P. Nov 5 '18 at 18:33