几何测度论笔记

1Week 1

Minkowski dimension

定义 1.1.

1.	For any $E\subset {\mathbb{R}}^n$, $\mathrm{diam}E={\sup }_{x,y\in E}|x-y|$;
2.	If $E\subset {\cup }_{\alpha \in A}{E}_{\alpha }$, then $\{E_{\alpha }{\}}_{\alpha \in A}$ is a covering of $E$. If $\{E_{\alpha }{\}}_{\alpha \in A}$ satisfies further $\mathrm{diam}{E}_{\alpha }\le \delta$, then $\{E_{\alpha }{\}}_{\alpha \in A}$ is called a $\delta$-covering of $E$.

定义 1.2.

1.	Given a bdd set $K\subset {\mathbb{R}}^n$, let $N(K,\delta )$ be the minimal cardinality of a $\delta$-covering of $K$, i.e. $$N(K,\delta )=\min \{J:K\subset \bigcup _{j=1}^J,\mathrm{diam}(E_j)\le \delta \}$$
2.	The upper Minkowski dimension is defined as $${\overline{\dim }}_M(K)=\underset{\delta \to 0}{\mathrm{limsup}}\frac{\log N(K,\delta )}{\log \frac{1}{\delta }}$$
3.	The lower Minkowski dimension is defined as $${\underline{\dim }}_M(K)=\underset{\delta \to 0}{\mathrm{liminf}}\frac{\log N(K,\delta )}{\log \frac{1}{\delta }}$$
4.	If ${\overline{\dim }}_M(K)={\underline{\dim }}_M(K)$, then it is called the Minkowski dimension of $K$, and is denoted by ${\dim }_M(K)$.

注 1.3.

1.	${\dim }_M(K)=\alpha ⟺\forall ϵ>0,\exists {\delta }_0>0$, s.t.$\forall \delta \in (0,{\delta }_0)$ $${\delta }^{-(\alpha -ϵ)}\le N(K,\delta )\le {\delta }^{-(\alpha +ϵ)}$$
2.	${\underline{\dim }}_M(K)=\alpha ⟺\forall ϵ>0,\exists {\delta }_0>0$, s.t.$\forall \delta \in (0,{\delta }_0)$ $$\alpha -ϵ<\underset{\eta \in (0,\delta )}{\inf }\frac{\log N(K,\eta )}{\log \frac{1}{\eta }}<\alpha +ϵ$$
3.	If $E\subset F$, then ${\dim }_M(E)\le {\dim }_M(F)$.

引理 1.4.

1.	Let $N_B(E,\delta )$ be the minima size of a covering of E consisting of closed balls of radii $\delta$, then we can replace $N(K,\delta )$ by $N_B(K,\delta )$ in the definition of ${\dim }_M(K)$.
2.	In $N_B(E,\delta )$ we can replace balls of radii $\delta$ by axis-paralled boxes of length $\delta$.
3.	If $A\subset {\mathbb{R}}^n$ is bounded, then $${\overline{\dim }}_M(A)={\overline{\dim }}_M(\overline{A})$$$${\underline{\dim }}_M(A)={\underline{\dim }}_M(\overline{A})$$

例 1.5.

1.	If $E$ is a finite set, then ${\dim }_M(E)=0$.
2.	If $E=[0,1]$, then ${\dim }_M(E)=0$.
3.	If $E\subset {\mathbb{R}}^n$ with ${\dim }_M(E)<n$, then ${\mathcal{L}}^n(E)=0$.
4.	If $E=\{0\}\cup \{1,\frac{1}{2},\frac{1}{3},\cdots \}$, then ${\dim }_M(E)=\frac{1}{2}$.

注 1.6. Let $E={\bigcup }_{j=1}^{\infty }{E}_j$, ${\dim }_M(E)={\sup }_j{\dim }_M{E}_j$ may fail.

定义 1.7. Let $P(E,\delta )$ be the greatest # of disjoint $\delta$-balls with centers in $E$,i.e. $$P(E,\delta )=\max \{k:\exists \text{ disjoint balls }B(x_i,\delta ),i=1,\cdots ,k,\text{ s.t. }{x}_i\in E\}$$

引理 1.8. $N_B(E,2\delta )\le P(E,\delta )\le N_B(E,\frac{\delta }{2})$. Consequently,$${\dim }_M(E)=\underset{\delta \to 0}{\lim }\frac{\log P(E,\delta )}{\log \frac{1}{\delta }}$$if it exists.

引理 1.9. Let $\alpha >0$, if $\exists C_{\alpha }>0$, s.t. $|E_{\delta }|\ge C_{\alpha }{\delta }^{\alpha },\forall \delta >0$, where $E_{\delta }=\{x:d(x,E)<\delta \}$. Then ${\underline{\dim }}_M(E)\ge n-\alpha$.

2Week 2

Outer Measure

定义 2.1. Let X be a nonempty set, a mapping $\mu :2^X\to [0,\infty ]$ is called an outer measure (we call it measure for short) of $X$ if it satisfies:

1.	$\mu (\varnothing )=0$
2.	If $A\subset {\bigcup }_{k=1}^{\infty }{A}_k$, then $\mu (A)\le {\sum }_{k=1}^{\infty }\mu (A_k)$

定义 2.2. $A\subset X$ is $\mu$-measurable if $\mu (B)=\mu (A\cap B)+\mu (B\setminus A)$,$\forall B\subset X$.

定理 2.3. If $\mu$ is a measure on $X$, then the collection of all $\mu$-measurable subsets of $X$ is a $\sigma$-algebra.

命题 2.4. Let $\mu$ be a measure on $X$.

1.	If $A\subset B\subset X$, then $\mu (A)\le \mu (B)$.
2.	$A$ is $\mu$-measurable $⟺A^c$ is $\mu$-measurable.
3.	$\varnothing$ and $X$ are $\mu$-measurable.
4.	If $\mu (A)=0$, then $A$ is $\mu$-measurable.
5.	If $E\subset X$ and $A$ is $\mu$-measurable, then $A$ is $\mu {|}_E$-measurable.

命题 2.5. Let $\{A_k{\}}_{k=1}^{\infty }$ be a sequence of $\mu$-measurable sets.

1.	${\bigcup }_{k=1}^{\infty }{A}_k,{\bigcap }_{k=1}^{\infty }{A}_k$ are $\mu$-measurable sets.
2.	If $\{A_k{\}}_{k=1}^{\infty }$ are disjoint, then $\mu ({\bigcup }_{k=1}^{\infty }{A}_k)={\sum }_{k=1}^{\infty }\mu (A_k)$.
3.	If $A_1\subset \cdots \subset A_k\subset \cdots$, then $\mu ({\bigcup }_{k=1}^{\infty }{A}_k)={\lim }_{k\to \infty }\mu (A_k)$.
4.	If $A_1\supset \cdots \supset A_k\supset \cdots$, with $\mu (A_1)<\infty$, then $\mu ({\bigcap }_{k=1}^{\infty }{A}_k)={\lim }_{k\to \infty }\mu (A_k)$.

定理 2.6. (Caratheodory’s criterion)
Let $\mu$ be a measure in ${\mathbb{R}}^n$. If $\forall A,B\in {\mathbb{R}}^n$ satisfying $\mathrm{dist}(A,B)>0$, we have $\mu (A\cup B)=\mu (A)+\mu (B)$. Then $\mu$ is a Borel measure.

定义 2.7. (i) A measure $\mu$ on ${\mathbb{R}}^n$ is Borel regular if $\mu$ is Borel and satisfies: $\forall A\subset {\mathbb{R}}^n$, $\exists$ a Borel set $B$, s.t. $A\subset B$ and $\mu (A)=\mu (B)$.
(ii) $\mu$ is a Radon measure if it is a Borel measure and satisfies:

1.	$\mu (K)<\infty$, for any compact $K\subset {\mathbb{R}}^n$.
2.	$\mu (V)=\sup \{\mu (K):K\subset V,K\text{ compact}\}$, for any open $V\subset {\mathbb{R}}^n$.
3.	$\mu (A)=\{\mu (V):A\subset V,V\text{ open}\}$, for any $A\subset {\mathbb{R}}^n$.

定理 2.8. Let $\mu$ be a Borel regular measure on ${\mathbb{R}}^n$. Suppose $A\subset {\mathbb{R}}^n$ is $\mu$-measurable and $\mu (A)<\infty$. Then $\mu {|}_A$ is a Radon measure.

定理 2.9. Let $\mu$ be a Borel measure on ${\mathbb{R}}^n$ and let $B$ be a Borel set.
(i) If $\mu (B)<\infty$, then $\forall ϵ>0$, $\exists$ a closed set $C$, s.t. $C\subset B$ and $\mu (B\setminus C)<ϵ$.
(ii) If $\mu$ is Borel regular and $\mu$ is locally finite, then $\forall ϵ>0$,$\exists$ an open set $V$, s.t. $B\subset V$ and $\mu (V\setminus B)<ϵ$.

定义 2.10. A Borel regular measure $\mu$ on a metric space $X$ is called uniformly distributed if $$0<\mu (B(x,r))=\mu (B(y,r))<\infty$$for any $x,y\in X$ and $0<r<\infty$.

定理 2.11. Let $\mu$ and $\nu$ be uniformly distributed Radon measures on ${\mathbb{R}}^n$. Then $\exists c\in (0,\infty )$, s.t. $\mu =c\nu$.

3Week 3

Hausdorff dimension

定义 3.1. (i) Let $A\subset {\mathbb{R}}^n$,$0\le s<\infty$,$0<\delta \le \infty$, then we define $H_{\delta }^s(A)=\inf \{{\sum }_{j=1}^{\infty }(\mathrm{diam}(C_j){)}^s:A\subset {\bigcup }_{j=1}^{\infty }{C}_j,\mathrm{diam}(C_j)\le \delta \}$;
(ii) $H^s(A)={\lim }_{\delta \to 0}{H}_{\delta }^s(A)={\sup }_{\delta >0}{H}_{\delta }^s(A)$, which is called $s$-dimensinal Hausdorff measure of $A$.

定理 3.2. For $0\le s<\infty$, $H^s$ is a Borel regular measure on ${\mathbb{R}}^n$.

定理 3.3. (i) $H^s$ is a counting measure;
(ii) $H^s\equiv 0$ on ${\mathbb{R}}^n$ provided $s>n$;
(iii) $H^s(\lambda A)={\lambda }^s{H}^s(A)$, $\forall s>0,A\subset {\mathbb{R}}^n$;
(iv) $H^s(L(A))=H^s(A)$,$\forall$ affine isometry $L:{\mathbb{R}}^n\to {\mathbb{R}}^n,A\subset {\mathbb{R}}^n$;
(v) $H^n=c{\mathcal{L}}^n$ for some constant $c>0$.