The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential and Riemannian geometry, as well as calculus of variations. Statements ...

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27 views

Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$

I am reading Lebesgue Integration on Euclidean Space by Frank Jones. My question is specifically regarding Chapter 4, Section C titled "The Lebesgue Function Associated with a Cantor Set". The author ...
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2answers
22 views

Steiner symmetrization of Lebesgue measurable set

I'm reading a proof in "Evans / Gariepy: Measure theory and fine properties of functions" of the following statemant: Theorem: Let $A \subset \mathbb R^n$ be $\lambda^n$-measurable, $a \in \mathbb ...
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2answers
61 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of H’s and T’s. Let N denote the number of tosses until you see “TH” for the first time. For example, for the sequence HTTTTHHTHT, we needed N = 6 ...
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23 views

Understanding integral equation with locally integrable function

I'm reading a proof in a book and don't understand a crucial step. So we have $f \in L^1_{\text{loc}}(\mathbb R^n)$ and $\phi \in C_c(\mathbb R^n)$. Now $\mu$ is a Radon measure, such that $$ \mu(A) ...
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48 views

Bounding dimension of IFS

Given the IFS $\{\frac x {2+x},\frac 2 {2+x}\}$ ($0\le x \le 1$) with attractor K prove that $0.53<\dim_HK<0.8$ I thought using the results from my last question by saying ...
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98 views

Proving ineqalities for the similarity dimension

a. Let $K$ be the attractor of the IFS $\{f_1,\dots f_n\}$ which satisfies SSC (i.e $f_i(K)\cap f_j(K)=\emptyset\forall i\neq j$) where for all $i, c_i$ such that $ 1\le i\le n, \space ...
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24 views

Upper Minkowski content and finite Hausdorff measure

Does someone know an example of a set $E$ with positive finite $s$-Hausdorff measure, Minkowski dimension $s$, and infinite $s$-dimensional upper Minkowski content ? The $s$-dimensional upper ...
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18 views

Minkowski dimension behaviour of sets with positive finite Hausdorff measure.

It is (rather) well known that the set \begin{equation*} E=\{k^{-1},k\in\mathbb{N}^{*}\} \end{equation*} has box-dimension $1/2$ and Hausdorff dimension $0$. However $H^{0}(E)=|E|=+\infty$. Is it ...
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25 views

Sard's Theorem with different Measures

From what I can tell Sard's theorem is formulated in terms of the Lebesgue measure. Is there a form of Sard's theorem for more general measures (in particular, those which are not absolutely ...
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24 views

Define Radon measure as an integral

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon measure and $f \in L^1_{loc}(\mathbb R^n, \mu)$, $f \geq 0$ on $\mathbb R^n$. Now we define an outer measure $\nu: \mathbb R^n \to ...
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42 views

Prove carpet has positive Hausdorff measure in its dimension

Given $D\subset\{0,1,2,\dots n-1\}\times\{0,\dots,m-1\}$, let $$K(D)=\{\sum_{k=1}^\infty(a_kn^{-k},b_km^{-k}):(a_k,b_k)\in D\forall k\}.$$ Show that if $D$ has uniform horizontal fibers (i.e. the ...
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34 views

Co-Area formula in Riemannian geometry

I wonder if the following holds true: Let $z:[0,1]\times B^{n-1}_r(0)\to(M^n,g), (t,p)\mapsto z(t,p)$ be a diffeomorphism a.e. onto its image with respect to the Lebesgue measure on $A:=[0,1]\times ...
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1answer
28 views

Differentiation theorem for Radon measures

I have trouble to understand a detail in the proof of the following Theorem: Theorem: Let $\nu, \mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be outer Radon measures, such that $\nu \ll \mu$. Then ...
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18 views

Proving that $\dim_M{K_{\lambda,\gamma}}=\dim_HK_{\lambda,\gamma}$

Define $K_{\lambda,\gamma}$ to be the attractor of the IFS $$\bigg\{\bigg(\matrix {\lambda&0\\0&\gamma}\bigg)\bigg(\array{x\\y}\bigg),\bigg(\matrix ...
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2answers
39 views

A property of Radon Measures

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon-measure. This means that every Borel-set $B \subset \mathbb R^n$ is $\mu$-measurable, $\mu$ is Borel-regular, i.e. for every set $A ...
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20 views

$\iint |x-y|^{-t} \,d\mu\, d\mu < \infty$ iff $t<1$

Consider the measure space $([0,1], \mathcal{L}([0,1]), \mu)$, where $\mu$ is the restriction of the Lebesgue measure to the closed interval $[0,1]$. I wish to show $\iint |x-y|^{-t} \,d\mu\, d\mu ...
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21 views

Poincaré inequality by capacity estimate.

Let $(X,d,\mu)$ be a complete metric space with a doubling measure $\mu$. For any ball $\mu(B)< \infty$ and $\mu(A)=\sup\{\mu(K);K\subset A, K \text{ compact}\}$. For any two closed disjoint ...
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89 views

How we can explain this by drawing a suitable diagram?

When I go through the proof of Riesz-Markov Representation Theorem I could encounter a step of defining $A_k=K\cap f^{-1}((y_k,y_{k+1}))$ $U_k=V\cap f^{-1}((y_k,y_{k+1}))$ And the $A_k\in ...
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53 views

A counterintuitive construction: finely covering a horizontal strip with vertical bands

Let $Q:=[0,1]^2$ and suppose we have a horizontal strip $S=\{\alpha\le y\le\beta\}\cap Q$ inside it (with $0\le\alpha<\beta\le 1$). We want to cover $S$ in an accurate way with vertical bands: ...
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54 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial ...
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43 views

The level set of Lipschitz functions

Suppose $u$: $R^N\to R$ is lipschitz, then do we have a.e. level set of $u$ has Lipschitz boundary? Is this anything to do with Sard theorem? Sard theorem states that a.e. Level set of smooth ...
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Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
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54 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
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33 views

Proving that plane - cantor - set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$. Denote the orthogonal projection of the set from the ...
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1answer
29 views

Fractal Dimension of $C_{\frac{1}{3}}\times[0,1]$

I wonder what is the dimension of the fractal set given by the product of the unit interval $[0,1]$ by the thirds-cantor-set ($C_\frac{1}{3}=\bigcap_n C_n$ where $C_0=[0,1],C_1=[0,\frac 1 3]\cup[\frac ...
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33 views

Fubini type results for Hausdorf dimension?

Suppose that I have a stack of hyperplanes in Euclidean space $\mathbb{R}^n$, let's call each plane $P_{a, x}=\{y\in\mathbb{R}^n\mid \langle y, x\rangle=a\}$ Suppose that a measurable subset $A$ of ...
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24 views

Coarea formula for fractional dimension

The coarea formula states that any locally Lipschitz function (e.g. a $C^1$-function) $F:\mathbb{R}^N\to\mathbb{R}^n$ with $N\geq n$ satisfies $$\int_A JF(x) \mathrm{d}\mathcal{H}^N = ...
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34 views

A conjecture about lines and points in the plane

Let $ l_1, l_2, \ldots$ be an infinite sequence of lines in the plane, and let $(a_1, b_1), (a_2, b_2), \ldots$ be an infinite sequence of pairs of points such that $a_i, b_i \in l_i$ and $a_i \neq ...
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46 views

Is this c the same as that c?

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
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35 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same ...
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34 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
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19 views

Existence of Lebesgue measure in parametric function space

I am thinking about this question but I can not solve. The question is: Can I define Lebesgue measure in the space of parametric functions and if the answer is yes what is that Lebesgue measure? Could ...
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1answer
28 views

A question about the Integral geometry and geometric probability.

In the book: Integral Geometry and Geometric Probability, (p16-17), the author proved that the measure of randomly throwing three points P1, P2, and P3 on the plane such that the circumdisk and the ...
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1answer
28 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
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30 views

Hausdorff measure and Ahlfors regular set

Let $E \subset \mathbb{R}^n$ be a closed set, not reduced to a point, and let $\mu$ be a (positive Borel) measure supported on $E$. If there is a constant $C \ge 1$ such that $C^{-1} r^d ≤ μ(E \cap ...
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Ahlfors regular

Let F be a closed set of $R^n$. Let $\mu$ be a borel regular measure on $R^n$. Let there be a constant C>0 such that for all $x\in F$, all $r\in (0,1]$ and all $\lambda \geq 1$ with $\lambda r \leq ...
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30 views

Find $\dim_H \operatorname{proj}_\theta E$ where is the circular Cantor set in the complex plane

$\newcommand{\proj}{\operatorname{proj}}$ $\dim_H(\cdot)$ is the Hausdorff dimension of a set. Let us denote the line through the origin in $\mathbb{R}^2$ which makes an angle of $\theta$ with the ...
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47 views

Hausdorff Dimension for Brownian motion over [0,1]

I am trying to calculate Hausdorff dimension for the trajectory of Brownian motion over $[0,1]$. I read the book of Morters and Peres and know that the dimension will be $\frac{3}{2}$. I tried to use ...
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22 views

Crofton formula in higher dimension

In the plane, the Crofton formula states that for a rectifiable plane curve $\gamma$, we have $\int |line \cap \gamma| d\Omega_1=2\times length(\gamma)$ where $d\Omega_1$ is the ...
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38 views

Use of Banach-like Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified.
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36 views

Hausdorff measures and densities

I've been stuck on this one for a while now. It's problem 2.4 from Falconer's "The geometry of fractals" Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with ...
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Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
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Finding countable compact set s.t $\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$

Im trying to find a countable compact set such that $$\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$$ I tried thinking about Koch curve, sierpinskii gasket and carpet, Bedford-McMullen carpet and ...
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35 views

Showing that $\mathcal{H}^s$ is Borel regular (assuming we know already know that $\mathcal{H}^s$ is measure)

I am trying to show that $\mathcal{H}^s$ (s-dimensional Hausdorff measure) is Borel regular. I am using the defintion $\mathcal{H}^s_{\delta}(F)=inf \Bigg\{ \sum_{i=1}^{\infty}|V_i|^s : \{V_i\} \text{ ...
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1answer
39 views

Proving that a mass distribution has positive Lebesgue measure

I am confused in this proof about how we obtain $\int f(u) \, d\mu(u) = \int f(u)g(u) \, d\mu(u)$ and how Plancherels theorem has been applied in $(6.6)$. Furthermore, I cannot understand how if ...
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22 views

Understanding the following expression in an integration

I cannot understand how the parts of this proof circled in red are obtained.
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53 views

A question on Hausdorff measure

This question is part of a homework assignment. Considering the hausdorff measure $\mathcal{H}_{2}$ on $\mathbb{R}^{3}$, I need to compute the measure of the unit cube: $A = \{(x,y,z) \in ...
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23 views

The existence of a measure of finite energy implies a lower bound on Hausdorff dimension

What is the significance of $\mu(x)=0$ and the use of continuity this proof? I am not quite sure about the general direction in the second paragraph.
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A covering argument for metric Jacobian

Given a Lipschitz map between Carnot Groups $ f : G_1 \to G_2$, with homogeneous dilations $ \delta^1_s, \delta^2_s$, we have the almost everywhere Pansu derivative $ D_H f(x)(y) = \lim_{s\to 0} ...
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2answers
44 views

Estimating the missing points of a 3D point cloud

Consider a cloud of N points (forming a smooth 3D object), in which n points are missing. Also, consider that there is no prior knowledge about the original shape of the point cloud. The only ...