Hausdorff measure

stein의 real analysis를 일부 정리한 내용입니다.

Exterior $\alpha$-dimensional Hausdorff measure of $E$를 다음과 같이 정의합니다.

$$m^*_{\alpha} = \lim_{\delta \to 0} \inf \{ \sum_k (diam F_k)^{\alpha} : E \subset \cup^{\infty}_{k=1} F_k , diam F_k \leq \delta \} $$

글괴 

$$\mathcal{H}^{\delta}_{\alpha} (E) = \inf \{ \sum_k (diam F_k)^{\alpha} : E \subset \cup^{\infty}_{k=1} F_k , diam F_k \leq \delta \}$$

는 $\delta$가 감소함에 따라 커지는 것을 알 수 있습니다.

 

Property 8 If $m^*_{\alpha} (E) < \infty $ and $\beta > \alpha $ then $m^*_{\beta} (E) =0 $. Also, if $m^*(E)>0$ and $\beta < \alpha$, then $m^*_{\beta} (E) = \infty $.

If $dima F \leq \delta $ and $\beta>\alpha$, then

$$ (diam F)^{\beta} = (diam F)^{\beta-\alpha} (diam F)^{\alpha} \leq \delta^{\beta-\alpha} (diam F)^{\alpha} $$

Therefore,

$$\mathcal{H}^{\delta}_{\beta} \leq \delta^{\beta-\alpha} \mathcal{H}^{\delta}_{\alpha} \leq \delta^{\beta-\alpha} m^*_{\alpha} (E) $$

Since $ m^*_{\alpha} (E) < \infty$, as $\delta \to 0$ , we have $m^*_{\beta}(E) =0$

The revsere holds when $m^*_{\beta} (E) = \infty $.

Lemma 2.2 Suppose a function $f$ defined on a compact set $E$ satisfies a Lipschitz condition with exponent $\gamma$. Then

(i) $m_{\beta} (f(E)) \leq M^{\beta} m_{\alpha} (E)$ if $\beta = \alpha/\gamma$