# theorem about hausdorff dimension involving capacity theory (theorem in the classic book of Heinonen)

I am studying the proof of this theorem :

Theorem: Suppose that $1<p \leq n$ and $E$ is a set in $R^n$ of $p -$ capacity zero . Then the Hausdorff dimension is at most $n-p$.

Proof:

"Since $cap_p E = 0$, E is contained in a $G_{\delta}$ set of $p$ - capacity zero, and hence using Choquet capacitability theorem we may as well assume that $E$ is compact ."

(*)Further, it suffices to show that $\Lambda_s^{\infty} (E) = 0$ for all $s$ with $n-p < s \leq n$. To this end, fix an open ball $B$ containing $E$ and choose a sequence of functions $u_j \in C^{\infty}_{0}(B)$ , admissible for the condenser $(E,B)$ with

$$\displaystyle\int_{B} |\nabla u_j|^p \ dx \rightarrow 0$$ as $j \rightarrow \infty$.

Then

$$\Lambda_{s}^{\infty} (E) \leq \Lambda_{s}^{\infty} ( \{ y \in B : |u_j (y) > 1/2 |\}) \leq c \displaystyle\int_{B} |\nabla u_j|^p \ dx$$

where $c$ is independent of $j$ . Hence $\Lambda_{s}^{\infty} (E) = 0$ and the theorem is proved. Someone can help me understand the lines with " " ? (From (*) I understand the proof)

the book is : Nonlinear potential theory of degenerate elliptic equations

authors: Juha Heinonen, Tero Kilpelainen and Olli Martio.

The proof of theorem is in the page 46. the Choquet theorem is in the page 32

Thanks in advance =)

• What should we do to help you understand these lines? – 40 votes Jul 25 '13 at 22:50
• I will write the proof of the theorem. It's better. sorry for my mistake . – math student Jul 25 '13 at 23:01