用户: Solution/ 习题: 几何测度论

1Dimensions

The main references of this chapter are [Bishop,Peres; Fractals in Probability and Analysis,Chapter 1], [Evans,Gariepy; Measure Theory and fine properties of functions, Chapter 1 and 2], and [Mattila; Geometry of Sets and Measures in Euclidean Spaces–Fractals and Rectifiability, Chapter 3 and 4].

1.1

1.

Let $\mathcal{A}\subset 2^X$ and $\mu :\mathcal{A}\to [0,\infty ]$ be such that $\varnothing \in \mathcal{A},X\in \mathcal{A}$,and $\mu (\varnothing )=0$. For any set $A\subset X$, define $${\mu }^{\ast }(A)=\inf \{\sum _{j=1}^{\infty }\mu (E_j):E_j\in \mathcal{A}andA\subset {\cup }_{j=1}^{\infty }{E}_j\}.$$Show that ${\mu }^{\ast }$ is an outer measure.

1.2

1.	Let $\mu$ be a Borel regular measure on ${\mathbb{R}}^n$ satisfying $\mu (K)<\infty$ for any compact subset $K\subset {\mathbb{R}}^n$. Show that Property 1 implies Property 2. Property 1: If $\mu (B)<\infty$, then, for any $\epsilon >0$, there exists a closed set $C$ such that $$C\subset B,\mu (B\setminus C)<\epsilon .$$Property 2: If $B$ is a subset of ${\mathbb{R}}^n$, then, for any $\epsilon >0$, there exists an open set $U$ such that $$B\subset U,\mu (U\setminus B)<\epsilon .$$
2.	Let $0<r<\infty$. Show that ${\mathcal{H}}^n(B(x,r))\in (0,\infty )$. As a result, there exists $c\in (0,\infty )$ such that ${\mathcal{H}}^n=c{\mathcal{L}}^n$.
3.	If $f:{\mathbb{R}}^m\to {\mathbb{R}}^n$ is Lipschitz, $0\le s\le m$, and $A\subset {\mathbb{R}}^m$. Then $${\mathcal{H}}^s(A)\le \mathrm{Lip}(f{)}^s{\mathcal{H}}^s(A).$$In particular, $\dim (fA)\le \dim A.$

2Orthogonal projections

The main references of this chapter are [Folland; Real Analysis, Chapter 4 and Chapter 7], [Evans, Gariepy; Measure theory and fine properties of functions, Chapter 1], [Mattila; Geometry of Sets and Measures in Euclidean Spaces–Fractals and Rectifiability, Chapter 1 and 8], and [Hochman; Lectures on fractal geometry and dynamics, Chapter 7]

2.1

1.	Let $X$ be an LCH space, $K$ a compact subset of $X$, and ${\left\{U_i\right\}}_{i=1}^n$ an open covering of $K$. There is a partition of unity on $K$ subordinate to ${\left\{U_i\right\}}_{i=1}^n$ consisting of compactly supported functions.

2.2

1.	Let $\mu$ be an $\alpha$-Frostman measure on ${\mathbb{R}}^n$ with $\alpha >n-1$, and $B(0,R):=\{x\in {\mathbb{R}}^n:|x|\le R\}$. Show that $$\mu (\partial B(0,R))=0.$$
2.	Suppose that $\mu$ is an $\alpha$-Frostman measure. Let $\psi \in {\mathcal{C}}_c^{\infty }({\mathbb{R}}^n)$ satisfy $\psi \ge 0$ and $\int \psi =1$. Show that ${\mu }_{\epsilon }:={\psi }_{\epsilon }\ast \mu$ are $\alpha$-Frostman measures (uniformly).
3.	Suppose that $\mu$ is an $\alpha$-Frostman measure on $E$. Show that there exist two compact subsets $K_1$ and $K_2$ of $E$ with $\mathrm{dist}(K_1,K_2)>0$ such that ${\mu |}_{K_1}$ and ${\mu |}_{K_2}$ are $\alpha$-Frostman measures.

2.3

1.	Let $B\subset {\mathbb{R}}^n$ be a Borel set of $\sigma$-finite ${\mathcal{H}}^s$-measure. Prove that $${\mathcal{H}}^s(B)=\sup \{{\mathcal{H}}^s(K):K\subset B\text{ is a compact set with }{\mathcal{H}}^s(K)<\infty \}.$$
2.	Prove that $${\int }_{{\mathbb{R}}^n}|f(x){|}^{p}dx=p{\int }_0^{\infty }{t}^{p-1}\left|\left\{x\in {\mathbb{R}}^n:|f(x)|>t\right\}\right|dt.$$
3.	Let $\mu$ be the measure on ${\mathbb{R}}^n$ with density $g(x)$. Let $M_{\lambda }(x)=\lambda x$. Show that $(M_{\lambda }{)}_{\#}\mu$ has density $${\lambda }^{-n}g({\lambda }^{-1}x).$$

3Differentiation of measures

The main references of this chapter are [Evans, Gariepy; Measure theory and fine properties of functions, Chapter 1 ], [Mattila; Geometry of Sets and Measures in Euclidean Spaces–Fractals and Rectifiability, Chapter 2]

3.1

1.	Let $\mu$ and $\lambda$ be Radon measures on ${\mathbb{R}}^n$. Show that $D(\mu ,\lambda ,x)$ is a Borel function.
2.	Let $\mu$ and $\lambda$ be Radon measures on ${\mathbb{R}}^n$. Suppose that $\mu \ll \lambda$. Show that $$\int D(\mu ,\lambda ,x{)}^{2}d\lambda x=\int D(\mu ,\lambda ,x)d\mu x.$$
3.	Let $\mu$ and $\lambda$ be Radon measures on ${\mathbb{R}}^n$. (i) Suppose $\mu \ll \lambda$. Show that $${\int }_{B}D(\mu ,\lambda ,x)d\lambda x=\mu (B).$$ (ii) Find $\mu$ and $\lambda$ such that $${\int }_{B}D(\mu ,\lambda ,x)d\lambda x<\mu (B)$$for some $B$.

4Fourier Transforms

The main references of this chapter are [Folland; Real Analysis, Chapter 8], [Mattila; Fourier analysis and Hausdorff dimension, Chapter 3, Chapter 4, Chapter 5], and [Mattila 1987; Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets]

4.1

1.	A measure is called continuous if each point has measure zero. Suppose $\{{\mu }_N\},N=1,2\dots$, and $\nu$ are non-negative Radon measures on $\mathbb{R}$, and that $v$ is continuous. Assume that $\hat{\mu }(\xi )\to \hat{\nu }(\xi )$ as $N\to \infty$, for each $\xi \in \mathbb{R}$. Then ${\mu }_N((a,b))\to \mu ((a,b))$ for all $a<b$.
2.	[Steinhaus’s Theorem] Let $E$ be a measurable subset of $\mathbb{R}$ such that its Lebesgue measure is positive. Then $${\Delta }^{\prime }(E):=\{x-y:x,y\in E\}$$contains an open interval containing $0$.
3.	Show that there exist compact sets $A\subset {\mathbb{R}}^n$ with $\dim A=n$ such that $\Delta (E)$ contains no interval $[0,\epsilon ],\epsilon >0$.

4.2

1.	Let $\psi \in {\mathcal{C}}_c^{\infty }\left({\mathbb{R}}^n\right)$ satisfy $\int \psi =1$, $\mathrm{supp}\psi \subset B(0,1)$, and $\psi \ge 0$. Let $\mu \in$ $M({\mathbb{R}}^n)$, and ${\mu }_{\epsilon }:=\mu \ast {\psi }_{\epsilon }$. Show that ${\delta }_{{\mu }_{\epsilon }}$ converges to ${\delta }_{\mu }$ weakly.
2.	Let $\mu \in M({\mathbb{R}}^n)$. If there exists a constant $C$ such that, for some $s>0$, we have $$\mu (B(x,r))\ge C{r}^S\forall x\in \mathrm{supp}\mu ,0<r<1.$$Show that, for $R>1$, $${\int }_{B(0,R)}|\hat{\mu }(\xi ){|}^{2}d\xi \ge C{R}^{n-s}.$$

5Self–similar sets

The main references of this chapter are [Hochman; Lectures on fractal geometry and dynamics, Chapter 5], [Bishop, Peres; Fractals in Probability and Analysis, Chapter 2], [Mattila; Fourier analysis and Hausdorff dimension, Chapter 9], and [Shen; Lectures on Bernoulli Convolution]

5.1

1.	Let $0<\epsilon <\delta$. Show that $A^{(\epsilon )}\subset A^{(\delta )}$, where $A^{(\epsilon )}:=\{x\in X:d(x,A)<\epsilon \}$.
2.	Let $(X,d)$ be a metric space, and $E\subset X$. Then the following statements are equivalent. (i) $E$ is complete and totally bounded. (ii) Every sequence in $E$ has a subsequence convergent to a point in $E$. (iii) $E$ is compact.
3.	Let $\{{\ell }_j\}$ be a sequence of lines in ${\mathbb{R}}^2$ such that (i) ${\ell }_j\cap B(0,1)\ne \varnothing$ for any $j$, and (ii) ${\ell }_j^{\prime }:={\ell }_j\cap B(0,2)$ is convergent under the Hausdorff metric. Show that the limit of ${\ell }_j^{\prime }$ is of the form $\ell \cap B(0,2)$ for some line $\ell$.

5.2

1.

Let $\Lambda$ be a finite set. For $i,j\in {\Lambda }^{\mathbb{N}}$, we define $d(i,j)=2^{-N}$, where $N$ is the largest number such that $$i_1\cdots i_N=j_1\cdots j_N.$$Let ${\tau }_d$ be the topology on ${\Lambda }^{\mathbb{N}}$ induced by $d$, and $\tau$ the standard product topology on ${\Lambda }^{\mathbb{N}}$. Show that $\tau ={\tau }_d$.

5.3

1.	Show that the strong separation condition implies the open set condition.
2.	Suppose $\Phi ={\left\{{\phi }_i\right\}}_{i\in \Lambda }$ is an IFS of similitudes. For $i=(i_1{i}_2\cdots i_n)\in {\Lambda }^{\ast }$, we define $r_i=r_{i_1}{r}_{i_2}\cdots r_{i_n}$. Fix $r>0$, and define $S_r=\left\{i\in {\Lambda }^{\ast }:r_i<r\le r_i^{\prime }\right\}$. Show that $S_r$ is a section.
3.	Suppose $(X,\mathcal{A},\mu )$ is a measurable space, and we call $A\in \mathcal{A}$ a $\mu$-atom if $\mu (A)>0$, and for any $B\in \mathcal{A}$ with $B\subset A$ we have either $\mu (B)=0$ or $\mu (A\setminus B)=0$. Show that (i) Let $A$ be a $\mu$-atom. Then for any $B\in \mathcal{A}$ with $B\setminus A$ we have either $\mu (B)=\mu (A)$ or $\mu (B)=0$. (ii) When $X={\mathbb{R}}^n$, and $\mu$ is a Borel measure, show that an $\mu$-atom is a singleton $\{x\}$ with $\mu (\{x\})>0$.