几何测度论笔记

1Week 1

Minkowski dimension

定义 1.1.

1.	For any $E\subset {\mathbb{R}}^n$,$diamE={\sup }_{x,y\in E}|x-y|$;
2.	If$E\in {\cup }_{\alpha \in A}{E}_{\alpha }$,then $\{E_{\alpha }{\}}_{\alpha \in A}$is a covering of E.If $\{E_{\alpha }{\}}_{\alpha \in A}$ satisfies further $diam{E}_{\alpha }\le \delta$,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 $$N(K,\delta )=min\{J:K\subset \bigcup _{j=1}^J,diam(E_j)\le \delta \}$$
2.	The upper Minkowski dimension is defined as $${\overline{dim}}_M(K)=\underset{\delta \to 0}{\mathrm{limsup}}\frac{logN(K,\delta )}{log\frac{1}{\delta }}$$
3.	The lower Minkowski dimension is defined as $${\underline{dim}}_M(K)=\underset{\delta \to 0}{\mathrm{liminf}}\frac{logN(K,\delta )}{log\frac{1}{\delta }}$$
4.	If ${\overline{dim}}_M(K)={\underline{dim}}_M(K)$,then it is called the Minkowski dimension of $K$,and is denoted by $di{m}_M(K)$.

注 1.3.

1.	$di{m}_M(K)=\alpha ⟺\forall ϵ>0,\exists {\delta }_0>0$,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.$\forall \delta \in (0,{\delta }_0)$ $$\alpha -ϵ<\underset{\eta \in (0,\delta )}{\inf }\frac{logN(K,\eta )}{log\frac{1}{\eta }}<\alpha +ϵ$$
3.	If $E\subset F$,then $di{m}_M(E)\le di{m}_M(F)$.

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 $di{m}_M(K)$.
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 $di{m}_M(E)=0$.
2.	If $E=[0,1]$,then $di{m}_M(E)=0$.
3.	If $E\subset {\mathbb{R}}^n$ with $di{m}_M(E)<n$,then ${\mathcal{L}}^n(E)=0$.
4.	If $E=\{0\}\cup \{1,\frac{1}{2},\frac{1}{3},...\}$,then $di{m}_M(E)=\frac{1}{2}$.

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

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

引理 1.8. $N_B(E,2\delta )\le P(E,\delta )\le N_B(E,\frac{\delta }{2})$.Consequently,$$di{m}_M(E)=\underset{\delta \to 0}{\lim }\frac{logP(E,\delta )}{log\frac{1}{\delta }}$$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 $A_1\subset ...\subset A_k\subset ...$,then $\mu ({\bigcup }_{k=1}^{\infty }{A}_k)={\lim }_{k\to \infty }\mu (A_k)$.
4.	If $A_1\supset ...\supset A_k\supset ...$,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 $dist(A,B)>0$,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.	$\mu (K)<\infty$,$\forall$ compact $K\subset {\mathbb{R}}^n$.
2.	$\mu (V)=sup\{\mu (K):K\subset VKcompact\}$,$\forall$ open $V\subset {\mathbb{R}}^n$.
3.	$\mu (A)=inf\{\mu (V):A\subset VVopen\}$,$\forall A\subset {\mathbb{R}}^n$.

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 $$0<\mu (B(x,r))=\mu (B(y,r))<\infty$$$\forall x,y\in X,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 $H_{\delta }^s(A)=\inf \{{\sum }_{j=1}^{\infty }(diam(C_j){)}^s:A\subset {\bigcup }_{j=1}^{\infty }{C}_j,diam(C_j)\le \delta \}$;
(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 $c>0$.