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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Proof step in Rademacher's Theorem

In the proof of Rademacher's theorem, we assume that $f: \Bbb R^n \to \Bbb R$ is a Lipschitz function and $v \in \Bbb R^n$ is a vector with $\Vert v \Vert = 1$. Our aim is to show, that $$ \mathrm ...
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54 views

What is the definition of Compact rectifiable set?

In dimension $\mathbb R^N$, we say a set $S$ is $N-1$-rectifiable if there exists a countably many $C^1$ hyper surface $\Gamma_i$ so that $$ \mathcal H^{N-1}(S_u\setminus \bigcup\Gamma_i)=0 $$ Now I ...
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47 views

Structure theorem for locally bounded variation functions

I have some doubts reading Measure Theory and Fine Properties of Functions by Evans and Gariepy. In particular, they define the space of locally bounded variation functions $BV_{loc}(U)$ in $U\subset \...
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39 views

Total variation of a vector valued measure

If I have understood correctly, a vector valued measure $\mu$ is simply a vector of measures, that is $\mu=(\mu_1,\dots,\mu_n)$, where $\mu_i$ is a possibly signed measure on the measure space $(X,\...
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27 views

The minimal $C^1$ hypersurface

Let $S_0\subset \mathbb R^N$ be given, where we assume that $S_0$ is a connected $C^1$ hyper surface and $\mathcal H^{N-1}(S_0)<\infty$. Here $S_0$ may not be able to represented as a graph. (up to ...
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41 views

Integral similar to Lebesgue point theorem

Assume we are in $\mathbb R^N$ and $\Gamma$ is a ($N-1$)-rectifiable set with $\mathcal H^{N-1}(\Gamma)<\infty$ and $\mathcal H^{N-1}(\bar \Gamma\setminus \Gamma)=0$. Let $u\in C_c(\mathbb R^N)$ ...
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17 views

Can a subset of $N−1$ rectifiable set be partitioned into countably many connected pieces? [duplicate]

This is a follow up question regarding to the problem discussed here. Thanks so much to @Silvia Ghinassi's nice and simple explanation there! Ok, here is the updated question: Let $\Gamma\subset \...
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1answer
26 views

Can a $N-1$ rectifiable set be partitioned into countably many connected pieces?

Let $\Gamma\subset \mathbb R^N$ be a $N-1$ rectifiable curve such that $\mathcal H^{N-1}(\Gamma)<\infty$. I am wondering that would it be possible to partition it into countably many connection ...
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21 views

What does it mean “Distance between $k$-planes induced by the identification plane-projection matrix”?

I'm reading some parts of Functions of bounded variation and free discontinuity problems by Ambrosio, Fusco, Pallara. At the very beginning of page 82 there's written "Let $G_k$ be the complete ...
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124 views

Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...
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64 views

Singular points of one-to-one mapping on rectifiable set

Let $E$ be an $s$-rectifiable set in $\mathbb{R}^n$ of positive $s$-dimensional Hausdorff measure $H^s(E)>0$. The original question I have is: Can there exist a one-to-one Lipschitz function $f\...
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2answers
39 views

Hausdorff dimension of homeomorphic compact metric spaces

Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff dimensions? ...
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1answer
40 views

Find parametrization for a possible “ruled” surface in $\mathbb R^4$

Let us endow $\mathbb R^4$ with a group law $\cdot$ such that the dilations $\delta_\lambda:(\mathbb R^4,\cdot)\to (\mathbb R^4,\cdot), (x_1,x_2,x_3,x_4)\mapsto (\lambda x_1,\lambda x_2,\lambda^2 x_3,\...
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23 views

Integral equality including function with compact support

I don't understand a proof step in my measure theory book, maybe you can help me. We have a Radon (outer) measure $\mu: \mathcal P(\Bbb R^n) \to [0, \infty]$ and a $f \in L^1_{\text{loc}}(\Bbb R^n)$, ...
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58 views

Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = \sup_{\...
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38 views

doubling property on manifold

I am learning the Geometric Measure Theory, and curious about how to generalize the covering lemma on manifold. However I am stuck at the doubling property: Let $(X,d,\mu)$ be a metric space with ...
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69 views

Borel regular measure on a subset of $\mathbb{R}^3$

Let $f(x,y)$ be a positive, differentiable function defined in the unit disc $D\subset \mathbb{R}^2$ and let $S=\{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2<1, z=f(x,y)\}$ Given $A \subset S$,...
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75 views

Is a bijective projection function measure preserving?

A subspace with dimension strictly less than the dimension of vector space has (Lebesgue) $measure=0$. Let $V$ be a vector space with $dimension=n$. To show that some set $S$ in V is zero-measure, is ...
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1answer
51 views

Weak isoperimetric inequality for surfaces in $\mathbb R^3$

The well-known isoperimetric inequality in $\mathbb R^2$ states that for a $\Gamma \subset \mathbb R^2$ a curve (simple, closed, piecewise $C^1$) and $A$ the area of the bounded component of $\Gamma^c$...
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44 views

Theorem on Measurability-Preserving Maps?

How to tell whether you should bother reading this: Consider two well known facts about measurable functions: (i) the supremum of a sequence of measurable functions is measurable, (ii) if $f$ is ...
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43 views

When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
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129 views

Does every bounded Jordan measurable set have porous boundary?

Let $A\subset \mathbb{R}^n$ be a bounded Jordan measurable set. I wonder if its boundary $\partial A$ is necessarily porous. I know that $\partial A$ has Lebesgue measure zero. I also think that one ...
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55 views

Sobolev Multiplication theorem for Fibre bundles

Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a manifold admitting a ...
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85 views

Inequality between Hausdorff measure and spherical Hausdorff measure

I have a doubt on spherical Hausdorff measure. Given $k, \delta \in (0,\infty)$, the $\delta$-Hausdorff premeasure is defined for $E\subset \mathbb R^n$ as: $$\mathcal H^k_\delta(E):=\inf\{\sum_j\...
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14 views

Euclidean regular hypersurfaces VS. Regular hypersurfaces in Heisenberg group

Let's fix the setting we are working in: let $\mathbb H^n$ be the Heisenberg group, we denote its points $P$ as $P=(x_1,\dots, x_n,y_1,\dots,y_n,t)$, the Lie algebra as $X_i=\frac{\partial}{\partial ...
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25 views

Continuously differentiable functions in the Heisenberg group

I need some help with the following. We are in the Heisenberg group $\mathbb H^n$: we denote points $P\in \mathbb H^n$ as $P=(x_1,\dots,x_n,y_1,\dots,y_n,t)$ and let $X_j=\frac{\partial}{\partial x_j}+...
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56 views

staircase length in Whitney's flat norm and Jenny Harrison's natural norm

Can someone provide the complete calculation for the length of a staircase as it converges to a diagonal line in Euclidean space in a sequence in which the number of steps goes to infinity between two ...
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155 views

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 $\mathbb{...
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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
36 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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53 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, $\mu(T)=0$: $$ \varphi(K)=\varphi(...
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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 ...
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1answer
59 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 $$\mu_x:=\frac{d(\bullet,x)}{\mu(B(x,d(...
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1answer
49 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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43 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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81 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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42 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
44 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
36 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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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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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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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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1answer
53 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 $ \...
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1answer
64 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 t}\varphi(p,t)=...
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1answer
38 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 \stackrel{*}{\...
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1answer
47 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
46 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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1answer
29 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
19 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*} \...
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1answer
92 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 $J=[x,x+h]$, ...