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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Lipschitz space-filling maps

First, some preliminaries and context. Let $f \colon [0,1]\to[0,1]^2$ be a space-filling curve. If we put on $[0,1]$ and $[0,1]^2$ the standard Euclidean metrics induced by $\mathbb{R}$ and ...
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77 views

Boundary of closure of open set in $\mathbb R^2$ has measure zero

Consider problem 4 on day 2 of this exam. Suppose that $\mathcal O\subset \mathbb R^2$ is an open set with finite Lebesgue measure. Prove that the boundary of the closure of $\mathcal O$ has ...
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1answer
28 views

Deduce probability density function from integrals on $N$ bounded regions

I stumbled upon this problem. Suppose you have an unknown probability density function on the two dimensional real plane, $f(x,y)$ Suppose you have three distinct points $P_1, P_2, P_3$ on the plane ...
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37 views

Image of a Jordan compact set under a degenerate map

Briefly: Is the image of a Jordan compact set $K$ under a degenerate smooth map $\varphi$ equal to an image of a compact subset $T\subseteq K$ of zero measure, $\mu(T)=0$: $$ ...
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65 views

Is an $L^p$ function in an annulus $L^p$ restricted to almost all planes?

Let $n\geq3$ and consider the annulus-like domain $A=B(0,1)\setminus B(0,r)\subset\mathbb R^n$. Take any number $p\in[1,\infty]$. If $f\in L^p(A)$, is it true that $f|_{P\cap A}\in L^p(P\cap A)$ for ...
2
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1answer
41 views

Doubling measure and Riesz Potential

I am currently trying to solve some analysis exercises on metric spaces, but I cannot quite tackle on of them. The exercises read as follows: Define the measure ...
3
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1answer
41 views

The derivative of a measure

Let $\mu$, $\nu$ be two Radon Measure on $\mathbb{R}^n$. How can I prove that $D_{\mu}{\nu}=\lim_{r \to 0} \frac{\nu(B(x,r)}{\mu(B(x,r))}$ is in $L^1_{loc}(\mathbb{R}^n,\mu)$?
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32 views

Uniqueness of measure

I'm studying the Lebesgue-Besicovitch differentiation theorem (on the book "Sets of finite perimeter and geometrical variational problem- Maggi") but I'm not able to understand the following: Why the ...
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41 views

Vector valued measures

I can't understand what a vector valued measure is. In particular what does it mean to write $\nu (A)$ with A Borel measurable set?
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1answer
31 views

Property of Radon measure on $\mathbb{R}^n$

Let $\mu$ a Radon measure on $\mathbb{R}^n$, $C$ a bounded and measurable set and $F_i$ a countable family of closed ball. I also have that $\frac{\mu(C)}{\eta(n)}\le \mu (C \cap \bigcup \{ \bar{B} ...
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1answer
33 views

calculus of the measure of a $C^1 $ hypersurface

I have to prove that: $$\lim_{r \to 0} \frac {\mathcal H ^{n-1}(M \cap B(x,r))}{\omega_{n-1} r^{n-1}}=1, $$ where $\mathcal H ^{n-1}$ is the ($n-1$)-dimensional Hausdorff measure, $M$ is a $C^1$ ...
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1answer
28 views

Action of differential on multivectors, what is it called?

Let $\varphi:X\to Y$ be a smooth map between finite dimensional vector spaces over $\mathbb R$. Let us define its differential in a pont $a\in X$ as a (linear) map $d\varphi(a):X\to Y$ acting by ...
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1answer
19 views

Regularity of limit measure and prove that $|\mu_h|\stackrel{*}{\rightharpoonup}|\mu|$

I have some questions. First of all, let $\mu_h$ a sequence of Radon measures and suppose that $\mu_h$ weakly-converge to another measure $\mu$. Now, this limit measure $\mu$ is still Borel? Is it ...
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18 views

Does the coarea formula hold for smooth maps with gradient bounded below?

The coarea formula for hypersurfaces in $\mathbb R^n$ can be written in two following forms: $$ \int_{\mathbb R^n} g(x) |\nabla u(y)| dx = \int_{\mathbb R} \int_{u^{-1}(t)} g(y) d\mathscr H^{n-1}(y) ...
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25 views

Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...
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12 views

Prove that $\int_{\mathbb R^n} u_\epsilon \, divT=\int_F div(T*\rho_\epsilon). $

Let $F\subseteq \mathbb R^n$ be a set and let $\chi_F$ its indicator function, regularized with functions $\rho_\epsilon\in C^\infty_c(\mathbb R^n)$ such that $u_\epsilon:=\chi_F* \rho_\epsilon\to 0$ ...
2
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1answer
32 views

Any relationship between Hausdorff measures

Let $ S_1= ( [0,1], d_1 ) $ and $ S_2 = ( [0,1], d_2 ) $ be two metric spaces, where $ d_1 = |x - y|$ and $d_2 = (1/2^i) $ where binary expansion of x and y matches upto $ i^{th} $ coordinate. Let $ ...
3
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1answer
38 views

Mean Curvature Flow

Recently I am reading the mean curvature flow from the lecture notes of Carlo Mantegazza where I found that Under mean curvature flow given by$$\begin{cases}{\partial\over \partial ...
2
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1answer
34 views

Regularity of $\phi$ in order that $\int g_h \phi \,dx \to \phi(0)$

Define the sequence of functions $(g_h)_h$ where $$g_h(x):= h\, \chi_{[0,1/h]}(x)$$ and the sequence of measures $$(\mu_h(dx))_h:= g_h(x)\,dx.$$ We want to show that $\mu_h ...
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1answer
43 views

Show limit exists of quotient of measures

This is a Theorem from Mattila's Book Geometry of sets and measures in Euclidean spaces: Let $\mu$ and $\nu$ be uniformly distributed Borel regular measures on a separable metric space $X$. There ...
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1answer
28 views

On a proof of essential uniqueness of uniformly distributed measures

In the book 'Geometry of sets and measures in Euclidean space' by P. Mattila, theorem 3.4 states that Let $\mu$ and $\nu$ be uniformly distributed Borel regular measures on a seperable metric space ...
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12 views

Properties of finite difference approximation to perimeter $|E\Delta (E+\epsilon)|/\epsilon$.

Let $E\subset\mathbb{R}$ be a measurable set. If $E$ is a finite union of disjoint intervals, then $E$ is said to have finite perimeter and its perimeter $P(E)$ can be expressed as the limit $$ P(E) ...
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1answer
26 views

Existence of a measure where diameter of balls can be estimated by their measure

In my last post I asked the following question: Given a metric space $X$, let $B_r(x):=\{y\in X\mid d(x,y)\leq r\}$ be the closed balls. Fix a real number $p>0$. My question now is: Is there a ...
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1answer
17 views

Existence of a measure defined on balls

Given a metric space $X$, let $B_r(x):=\{y\in X\mid d(x,y)\leq r\}$ be the closed balls. Fix a real number $p>0$. My question now is: Is there a Borel-measure $\mu$ on $X$ such that \begin{align*} ...
2
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1answer
48 views

Prove Property of Doubling Measure on $\mathbb{R}$

Problem. Let $\mu$ be a fixed finite measure on $\mathbb{R}$. $\mu$ is said to be doubling if there exists a constant $C>0$ such that for any two adjacent intervals $I=[x-h,x]$ and ...
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96 views

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 ...
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1answer
25 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 ...
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26 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
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1answer
40 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 $\alpha\geq 1$ ...
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2answers
36 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
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1answer
42 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 $\frac 1 3$ ...
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23 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
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2answers
164 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 ...
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23 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 ...
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22 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 ...
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3answers
64 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 ...
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1answer
87 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 $C^1$ ...
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2answers
577 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 ...
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1answer
29 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 ...
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27 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 ...
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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 ...
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78 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 ...
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32 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 ...
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1answer
56 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 ...
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1answer
52 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 $H^d$ ...
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1answer
33 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 ...
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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^+} ...
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29 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 ...
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2answers
67 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) ...
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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 ...