- For a Radon measure \(\mu ,\) with \(\textstyle \int _ { \mathbb { R } ^ { n } } d \mu = C\) . Prove that for all \(\epsilon > 0\) , there exists a set \(E _ { \epsilon } \subset \mathbb { R } ^ { n }\) such that \[ \mathcal M^1(E_\epsilon) :=\inf_{E_\epsilon\subset\bigcup_i B_i} \sum_i\operatorname{diam}(B_i) <10\epsilon, \] and for any \(x\notin E_\epsilon\) and \(r>0\),
\[ \int _ { B _ { r } ( x ) } d \mu \leq \frac { C r } { \epsilon } . \]
(Hint: use Vitali covering lemma.)
Solution
1. Define the exceptional set by failure of the desired ball estimate.
Set \[ E_\varepsilon := \left\{x\in\mathbb R^n: \text{there exists }r>0\text{ with } \mu(B_r(x))>\frac{Cr}{\varepsilon} \right\}. \] Then by definition, if \(x\notin E_\varepsilon\), we have \[ \mu(B_r(x))\le \frac{Cr}{\varepsilon} \] for every \(r>0\).
2. Observe that every bad witnessing ball has small radius.
If \(x\in E_\varepsilon\) and \(B_r(x)\) witnesses membership, then \[ \frac{Cr}{\varepsilon}<\mu(B_r(x))\le\mu(\mathbb R^n)=C. \] If \(C>0\), this implies \[ r<\varepsilon. \] If \(C=0\), the conclusion is trivial with \(E_\varepsilon=\varnothing\). Thus all witnessing radii are uniformly bounded.
3. Apply the Vitali \(5r\) covering lemma.
Consider the family of all witnessing balls \[ \mathcal B =\left\{B_r(x):x\in E_\varepsilon, \ \mu(B_r(x))>\frac{Cr}{\varepsilon}\right\}. \] By the \(5r\) covering lemma, there exists a countable pairwise disjoint subfamily \[ B_{r_i}(x_i) \] such that \[ E_\varepsilon\subseteq\bigcup_i B_{5r_i}(x_i). \] Since the selected balls are disjoint, \[ \sum_i\mu(B_{r_i}(x_i)) \le \mu(\mathbb R^n)=C. \] But every selected ball is bad, so \[ \mu(B_{r_i}(x_i))>\frac{Cr_i}{\varepsilon}. \] Therefore \[ \sum_i r_i <\frac{\varepsilon}{C} \sum_i\mu(B_{r_i}(x_i)) \le\varepsilon. \]
4. Estimate the Hausdorff \(1\)-content.
The balls \(B_{5r_i}(x_i)\) cover \(E_\varepsilon\), and \[ \operatorname{diam}(B_{5r_i}(x_i))=10r_i. \] Hence \[ \mathcal M^1(E_\varepsilon) \le\sum_i10r_i <10\varepsilon. \] Together with Step 1, this gives the required exceptional set and ball estimate.