1

I'm reading Mattila's book "Geometry of sets and measures in Euclidean spaces". At the p. 222 in proof of Theorem 16.2 we have the following proposition:

Let $\varepsilon >0$. Since E ($\mathcal H^m(E) < \infty$) has positive lower density ($\Theta^m_*(E, \cdot) > 0$) $\mathcal H^m$ a.e. on $E$, there are a compact subset $F$ of $E$ with $\mathcal H^m(E \setminus F) < \varepsilon$ and positive numbers $\delta$ and $r_0$ such that \begin{equation}\label{} \mathcal H^m \big( E \cap B(a,r)\big) > \delta r^m \tag{$\star$} \end{equation} for $a \in F$, $0 < r < r_0$.

I know that for a.e. point $a \in E$ there exists a $r_0=r_0(a)$ and $\delta=\delta(a)$ such that condition $(\star)$ holds. Also, Borel regular measures (like a $\mathcal H^m$ in $\mathbb R^n$) can be approximated in the next way: if $\mathcal H^m(A) < \infty$ and $\varepsilon >0$ there a closed set $C$ s.t. $\mathcal H^m(A \setminus C) < \varepsilon$.

I can't understand how to find such set $F$.

Danil
  • 116
  • 4

1 Answers1

1

I found an answer to my question. Here we will use following properties of $\Theta^m_*$:

  1. $\Theta^m_*$ is a Borel function,

  2. $\Theta^m_*(E, x) = \lim\limits_{\delta\to 0} \inf\limits_{r < \delta} r^{-m} \mathcal H^m\big(E \cap B(x,r)\big)$ and $\inf\limits_{r < \delta} r^{-m} \mathcal H^m\big(E \cap B(x,r)\big)$ is Borel function too.

Since $\mathcal H^m(E)<\infty$ we can find $\mathcal H^m$-measurable set $B\subset E$ with $\mathcal H^m(E\setminus B) < \varepsilon$ and $$ \inf\limits_{r < 2^{-k}} r^{-m} \mathcal H^m\big(E \cap B(x,r)\big)\rightrightarrows \Theta^m_*(E, x) \quad\text{for $x\in B$} \tag{1}. $$ It's Egorov's theorem.

Again $\mathcal H^m(B) < \infty$ and $\Theta^m_*$ is a measurable function. By Lusin's theorem we obtain that there exists a compact set $F$ such that $\mathcal H^m(B\setminus F) < \varepsilon$ and $\left. \Theta^m_*(E, \cdot)\right|_F$ is a continuous function and $\Theta^m_*(E,x) \ge \eta$ for some $\eta >0$ and for all $x\in F$. That's why we can find a $r_0$ such that $$ \inf\limits_{r < r_0} r^{-m} \mathcal H^m\big(E \cap B(x,r)\big) > \Theta^m_*(E,x) - \eta/2\ge \eta/2 \quad\text{ for all $x\in F$}. $$

Danil
  • 116
  • 4