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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3
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54 views
+100

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to ...
0
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0answers
50 views

What is the extent of the streaks covering a square? [closed]

Let $N$ be a unit square, $1 < A <\sqrt{2}$ real number, we put a strip with width $A$ to the square randomly. I would like to determine the measure of the strips, that cover the square. I don't ...
0
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1answer
23 views

Inequality with supremum of divergence of vector fields

I would like to write this inequality: $$\sup\left\{ \int_{A \cup (\mathbb{R}^d \setminus F)} \operatorname{div} T \space \rm d m^d : T \in C_0^1(\mathbb{R}^d,\mathbb{R}^d), |T(x)| \le 1 \right\} \le ...
2
votes
0answers
16 views

Blow-up of derivative of BV function at the jump set

"Motivation" Let $u\in BV(\mathbb{R}^n)$ be a function of bounded variation, and let $x\in J_u$ be a point in its jump set. For $\mathcal{H}^{n-1}$-a.e. such $x$, we can define the unit normal $\nu$ ...
2
votes
1answer
33 views

Proving a Certain Planar Measure Is Zero on Horizontal Lines

Question: Suppose $\mu$ is a measure on $\mathbb{R}^{2}$ with respect to which all open squares are measurable. Suppose $\mu$ has the following property: there exists a constant $\alpha\geq 1$ ...
1
vote
2answers
35 views

Hausdorff measure of a subset of $\mathbb R^3$

Let $f \in L^1_{\text{loc}}(\mathbb R^3)$. We define $A \subset \mathbb R^3$ as $$ A := \left\{ x \in \mathbb R^3 \, : \, \limsup_{r \to 0} \frac 1 r \int_{\mathbb B(x,r)} \vert f(y) \vert \, \mathrm ...
2
votes
1answer
32 views

Hausdorff dimension of a Sierpinski-like triangle

Define the set $A \subset \mathbb R^2$ by proceeding as follows. Let $A_0$ be a closed equilateral triangular region of side 1. $A_1$ are the three equilateral triangular regions of side $\frac 1 3$ ...
0
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0answers
21 views

A property on exterior measures on a metric space

Let $(X,d)$ be a metric space and $\mu:2^X\to[0,\infty]$ be an exterior measure on $X$, $\mathcal{M}:=\{E\subset X:\forall S\subset X\ \ \ \mu(S)=\mu(S\cap E)+\mu(S-E)\}$. Suppose ...
3
votes
2answers
139 views

Is there a well-defined notion of measure zero on topological manifolds?

We extend the concept of measure zero on manifolds by local parameterization. but in this definition we have to check if it is true for every parametrization. In Guillemin's Differential Topology this ...
0
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0answers
21 views

Weak Compactness Therem for $L^p$

I have a problem to understand a point in the Weak Compactness Theorem from the book of Evans/Gariepy. So we have $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ a Radon measure with $\mu \ll ...
0
votes
0answers
18 views

Reconstructing a measure from its (absolutely continuous) marginals

Let's denote by $C$ the space of continuous functions $[0,T] \rightarrow \mathbb{R}^n$ for some fixed $T>0$ and assume we have a probability measure $Q$ on the space $C$. Consider the evaluation ...
0
votes
3answers
54 views

Can I cover a square with many line segments?

Not sure If I've chosen the tags correctly. Anyway, is it possible to obtain a unit square with enough line segments oriented vertically, placed next to each other? We know that a unit square has ...
3
votes
1answer
79 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
30
votes
2answers
548 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
1
vote
1answer
28 views

Differences in defining the packing (outer) measure

The definition of a packing measure in Falconer's Fractal geometry is given by I am assuming that $\mathcal{P}^s(F)$ as defined in 3.24 is an outer measure (this is not stated in the book). Now ...
0
votes
0answers
23 views

Vaguely relative compact subsets of Radon measures on $\sigma$-compact Polish spaces

Let $ (X, d) $ denote a Polish space and let $M(X)$ denote the space consisting of all finite signed Radon measures on $(X,d)$. We say that a sequence $ \{ \mu_k \}_{k\geq 1} \subset M(X) $ converges ...
4
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0answers
25 views

Wasserstein space of order 2.

I have a question about Wasserstein space. I am just wondering if the following statement is true Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with $\bar{\Omega}$ being a Polish space ...
4
votes
0answers
51 views

Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
0
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0answers
29 views

Slicing rectifiable sets with maps into the sphere

Notation and Context (Might not be helpful) Let $S\subset\mathbb{R}^n$ be a countably $\mathcal{H}^{M}$-rectifiable set. If $f:\mathbb{R}^n\to\mathbb{R}^k$ is a Lipschitz function, then we can find ...
-1
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1answer
46 views

Prove the set is Jordan- measurable and find the appropriate Jordan measure (volume) of the set V.

Prove the set is Jordan- measurable and find the appropriate Jordan measure (volume) of the set V. $$V= \{ (x,y,z)|x^2+y^2\geq 1, x^2+y^2\leq2x, x^2+y^2+z^2 \geq 4, x+y+z\leq 8\} $$ I'm not sure if ...
1
vote
1answer
32 views

Image of $A\subset \mathbb R^d$ under a Lipschitz function is $H^d$ measurable.

I really need an help with the following exercise. Suppose that $A\subseteq \mathbb R^d$ is Lebesgue measurable. Let $f\colon A \to \mathbb R^k $ be a Lipschitz function. Show that $f(A)$ is $H^d$ ...
1
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1answer
26 views

Hausdorff measure of $f(A)$ where $f$ is a Holder continuous function.

Let $f\colon \mathbb R^d\to \mathbb R^k$ be a $\beta-$ Holder continuous function ($\beta \in (0,1)$) and $A\subset \mathbb R^d$. As for a Lipschitz function $g$ it holds that $H^s(g(A))\leq Lip(g)^s ...
1
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1answer
15 views

$H^s(A)=0 \iff H^s_\infty(A)=0$

I have the following exercise: Show that $H^s(A)=0 \iff H^s_\infty(A)=0$ for $A\subset \mathbb R^d$. Here $H^s(A)$ is he Hausdorff measure of the set $A$, so $H^s(A):=\lim_{\delta\to 0^+} ...
1
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0answers
21 views

Does the coarea formula hold for delta-function?

Let $\Omega \subset \mathbb R^n$ be an open bounded domain, $u \colon \Omega \to \mathbb R$ be a Lipshitz function and suppose that $\nabla u (x) \neq 0$ for $x \in \Omega$. The coarea formula tells ...
0
votes
2answers
66 views

Bounded bessel functions in an s-set projection proof

The following is an extract from Falconer's Geometry of Fractal Sets about the proof of: "...Using the definition of a Bessel function $J_0=\frac{1}{2\pi}\int^{2\pi}_0 \cos(u \cos \theta) ...
0
votes
1answer
19 views

Easy argument in Lemma of Corea formula

I don't understand a presumably easy argument in my textbook. Let $L: \mathbb R^n \to \mathbb R^m$ be a linear map, $n \geq m$, $A \subset \mathbb R^n$ $\lambda^n$-measurable. We assume that $\dim ...
1
vote
1answer
44 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 ...
0
votes
2answers
38 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 ...
2
votes
2answers
118 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 ...
0
votes
1answer
27 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) ...
1
vote
1answer
58 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 ...
0
votes
1answer
109 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 ...
0
votes
1answer
34 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 ...
1
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1answer
34 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 ...
0
votes
0answers
35 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 ...
1
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0answers
34 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 ...
1
vote
1answer
47 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 ...
4
votes
0answers
52 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 ...
1
vote
1answer
36 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 ...
1
vote
2answers
48 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 ...
2
votes
1answer
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 ...
0
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0answers
32 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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0answers
129 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 ...
0
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1answer
56 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: ...
2
votes
0answers
90 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 ...
0
votes
0answers
76 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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votes
1answer
62 views

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 ...
1
vote
0answers
61 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 ...
1
vote
1answer
37 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 ...
3
votes
1answer
37 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 ...