Other references for Marstrand's density theorem I am currently working on topics closely correlated with Marstrand's density theorem (If the measure has a density, then the exponent of the density is an integer). I was reading together Preiss's article and De Lellis's notes but I realised that the proof of Mastrand theorem in De Lellis's note is wrong:
The mistake is that at pg 27 the baricenter $b(r)$ cannot pass to blowup, since it has not been normalized properly. Does anyone know another (modern enough) reference which I can use to read the proof? Marstrand's paper uses quite an obsolete notation and it is very hard to follow.
Thank you in advance
 A: What a curious coincidence! I was reading Marstrand's theorem from De Lellis' notes just a couple of weeks ago (after reading it a year ago without checking all the details) and I noticed the same mistake!
You can fix it in the following way: first, the right scaling is 
$$c(\rho)=\lim_{i\to\infty}\frac{b(\rho r_i)}{r_i}\qquad\text{and}\qquad z=\lim_{i\to\infty} \frac{y_i}{r_i}$$
Now the idea is, instead of computing $\langle \frac{b(\rho r_i)}{r_i},\frac{y_i}{r_i}\rangle$, to compute $\langle \frac{b(\rho r_i)}{r_i},\frac{y_j}{r_j}\rangle $ with $r_j$ much smaller than $r_i$. 
More precisely choose a subsequence $r_{j(i)}$ such that 
$$\frac{r_{j(i)}}{r_i}\to 0\quad\text{as} \quad i\to\infty.$$
In this way
\begin{align}
|\langle c(\rho),z\rangle |&=\lim_{i\to \infty}\left|\left\langle \frac{b(\rho r_i)}{r_i},\frac{y_{j(i)}}{r_{j(i)}}\right\rangle\right|\\
&=\lim_{i\to \infty}\frac{1}{r_ir_{j(i)}}|\langle b(\rho r_i),y_{j(i)}\rangle|\\
&\leq \lim_{i\to \infty}\frac{1}{r_i r_{j(i)}} C(\alpha)\|y_{j(i)}\|^2\\
&=C(\alpha) \lim_{i\to\infty}\frac{\|y_{j(i)}\|}{r_{j(i)}}\frac{\|y_{j(i)}\|}{r_i}=0
\end{align}
where we used the quadratic decay of the barycenter and the fact that $y_{j(i)}\in B_{2\rho r_{j(i)}}\subset B_{2\rho r_i}$.
A: Pertti Mattila's textbook Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability contains a proof of Marstrand's density theorem (it's theorem 14.10 from that book). I did not try to read it, though.
