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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1answer
246 views

Haar measure on the groups SO(n) and SO(n,m)

Would you please give me some information about Haar measure on special orthogonal group SO(n) and indefinite special orthogonal group SO(n,m)? Thank you so much!
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1answer
86 views

Hausdorff dimensions of smooth but non-rectifiable curves

Smooth curves of finite length have a Hausdorff dimension of 1. How about smooth but non-rectifiable (i.e. infinite-length) curves? Are they also of Hausdorff dimension 1, or does it depend on the ...
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0answers
23 views

I cannot check $\mathcal{A}_0(X)$ is an algebra.

I'm reading introduction to the theory of currents written by Dinh.I'm confused in proposition 1.1.3. Say,$X$ is a unit disc in $\mathbb{R}^2$,$A:=[-1/2,1/2]\times [-1/2,1/2]\subset X$,then ...
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1answer
42 views

Is there a rational number describing the ratio of a volume, as a string, to a surface area?

If you were to take an arbitrary 3-dimensional shape with finite surface area, then look at the volume of that shape, turn the volume into a long cylindrical string bunched up ideally inside the shape ...
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1answer
29 views

An exercise about Lebesgue measure.

let $\{E_{n}\}_{n\in\Bbb N}$ be a sequence of measurable subsets such that $E_n \subseteq (0,1)$and let $\limsup_{n\to\infty} m(E_{n})=1$, prove that exists a subsequence {$E_{n_{k}}$} such that ...
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58 views

Linearising angle to chord length over a reasonable domain of < pi; or, how to make measuring a rock with a protractor easy

Imagine that I have a protractor and compass, and wish to use it to measure the distance between two points (potentially in three dimensional space, such as on a rock). However also being a forgetful ...
3
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1answer
51 views

Any two polish atomless probability spaces are isomorphic

On page 7 of Villani's Optimal Transport: Old and New (page 19 in this preprint), he states that any two polish atomless probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ are ...
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0answers
45 views

Isometric trivialization of tangent bundle of Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold. $x\in M$. We know that there is a neighborhood $U$ of $x$ we can have a isometric trivialization $\pi$ of $TM$, Given by the orthonormal frame on $U$. Formally, ...
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1answer
37 views

2-dimensional density of a cone (by Morgan's Geom. Measure Theory)

I was reading Morgan's "GMT: a beginner's guide" and I stucked on a very simple fact. I was not very familiar with Hausdorff measures, hence I have some troubles. The topic is the same of this ...
3
votes
2answers
77 views

Separability almost everywhere

Let $(X,d)$ be a metric space and $\mu$ a Borel probability measure. Suppose that for every $\epsilon>0$ we have that $\mu(B_{\epsilon }(x))=c_{\epsilon}>0$ a.e. Is this enough to show that ...
2
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0answers
59 views

Measures whose projections are absolutely continuous

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely ...
3
votes
1answer
152 views

Is measure $m$ invariant under change of metric?

$\newcommand{\vol}{\operatorname{vol}}$ Let $(M^n, g)$ be a Riemannian manifold. We define $d\vol(g)$ as a canonical volume form and then integrate. It is clear that $d\vol(g)$ depends on the metric ...
0
votes
1answer
56 views

a measurable subset of the plane with restricted sections

Is there a bounded subset $X\subset \mathbb R^2$ of positive measure such that for almost every line $L\subset \mathbb R^2$ the $1$-dimensional Hausdorff measure of $X\cap L$ is either 0 or 1? If ...
2
votes
3answers
100 views

Online reference about Geometric Measure Theory.

I would like to find an online reference about the basics of Geometric Measure Theory. The reference should treat such things as regions and isoperimetric surfaces. Can you tell me, where I can find ...
1
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1answer
45 views

Does the squared root $\sqrt{|\cdot|}$ belong to $BV((-1,1))$?

Does the squared root belong $\sqrt{|\cdot|}$ to $BV((-1,1))$? In an affirmative case, what is its derivative in the distributional sense, i.e., what is the Radon measure $\mu$ such that $\mu=Du$ ?
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1answer
103 views

Two Definitions of Minkowski Dimension

I'm currently reading a paper. Let $F\subset\mathbb R^n$ and $\epsilon\gt0$, the paper defined $m^s(F):=\liminf_{\epsilon \to 0}\epsilon^{s-n}\lambda(F_\epsilon)$ and $M^s(F):=\limsup_{\epsilon \to ...
1
vote
1answer
138 views

Simple questions about Hausdorff measure

Let $s>0$ and $0 < \delta \leq \infty$. For a set $E \subset R^n$ define $$ H_{s}^{\delta} (E)=\inf \left\{ \sum_i r_{i}^{s}\right\},$$ where the infumum is taken over all coverings of $E$ by ...
1
vote
1answer
42 views

Uniform Rectifiability

What is the definition of uniform rectifiability as used in the context of analytic capacity of compact sets in $\mathbb{C}$? The precise context is this paper by Mattila, Mernikov and Verdera.
3
votes
1answer
101 views

Non-trivial Hausdorff measures for general metric spaces

For a metric space $(X,d)$ and a dimension function $\varphi:[0,\infty)\to[0,\infty)$ we can define a metric outer measure $H_\varphi$ which is $\varphi$-Hausdorff measure. Since it is a metric outer ...
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vote
1answer
40 views

approximate tangent planes and standard tangent planes

Let $ h: U \rightarrow R^{n-m} $ be a $ C^1 $-function on $ \overline{U} $, where $ 0 \in U \subset R^m $ is an open set. Suppose $ h(0)=0 $ and $ \nabla h (0)=0 $. Then if $ M \subset R^n $ is the ...
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1answer
158 views

metric and measure on the projective space

Let $RP^n$ be the $n$-dim real projective space. I have the following four questions. What is the so called standard metric on $RP^n$? More generally, consider a metric space $M$ with an equivalent ...
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0answers
46 views

Meaningful measures for comparing infinite dimensional geometric objects

I have two infinite-dimensional convex polytopes, call them $A$ and $B$. I know that $B$ is completely contained within $A$, and I want to say something meaningful about their relative sizes. From ...
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1answer
50 views

Hausdorff measure of an embedded submanifold

Let $ M $ be a $ C^1 $- embedded n-submanifold (without boundary) of $ R^{n+k} $. Is it true that for every $ K $ compact set in $ R^{n+k} $ the n-dimensional Haussdorf measure of $ M \cap K $ is ...
4
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0answers
172 views

Lebesgue measure as a fixpoint: change of variables formulas

This question is inspired by several others on a similar topic: see e.g. this one and a sequence of linked questions. Let us so far focus on $\Bbb R^n$ endowed with standard Borel structure. For any ...
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vote
1answer
49 views

Density of an $n$-dimensional embedded submanifold

Let $ M $ be a n-dimensional embedded $ C^1 $ submanifold in $ R^{n+k} $. Let $ H^n $ be the n-dimensional Haussdorf measure of $ R^{n+k} $. I want to prove that $$ \lim_{r \rightarrow 0} ...
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vote
1answer
97 views

change of variable for Hausdorff measure

Let $U$ be an open set in $\mathbb{R}^n$, $T:U \rightarrow \mathbb{R}^n$ be a diffeomorphism defined on $U$, $\mu$ be the $d$-dimensional Hausdorff measure, $0<d<n$. Is it true that we have the ...
4
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0answers
100 views

Alternative rigorous definition of a surface integral

Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
2
votes
1answer
134 views

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
4
votes
2answers
99 views

Is the validity of measuring area by approximation an assumption of calculus?

The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
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0answers
48 views

Approximate tangent planes and densities

I'm studying the proof of the following theorem (Simon 'Lectures on Geometric measure theory' Theorem 11.6): Theorem Let $ M \subset R^{n+k} $ be $ H^n $-measurable (where $ H^n $ is the Haussdorf ...
6
votes
1answer
148 views

Homogenous measure on the positive real halfline

Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where ...
2
votes
1answer
175 views

The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$

I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
1
vote
1answer
303 views

Riemannian measure and Hausdorff measure in a general Riemannian Manifold

Let $ M $ be a Riemannian manifold and let $ \mu $ be its Riemannian measure. This is the measure obtained by Riesz reprersentation theorem such that for every continuous function with compact support ...
0
votes
1answer
179 views

Radon-Nikodým derivative with respect to the Lebesgue/Hausdorff measure is always defined

Question 1. Is it possible to say that the Radon-Nikodým derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to the Lebesgue measure is always defined but may be a generalized ...
2
votes
1answer
109 views

A question about the proof of Rademacher theorem

I'm referring to the proof of Rademacher theorem due to C.B.Morrey (i'm reading it on Simon: 'Lectures on geometric measure theory').\ The proof can be summarized in the following steps:\ 1)For every ...
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0answers
40 views

A question about Lebesgue measure 3

Let $ L^k $ be the k-dimensional lebesgue measure. Let $ A \subset R^n $ be a Borel set. Suppose we have proved that $ L^1(A \cap l )=0 $ for each line $ l $ parallel to some line passing throught the ...
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0answers
48 views

A question about Hausdorff measure

Let $ \mathscr{H}^m $ be the m-dimensional Hausdorff measure in $ R^n $. Let $ m<k $ and $ A \subset R^n $. If $ \mathscr{H}^m(A) < \infty $ then $ \mathscr{H}^k (A) =0 $. How can i prove this ...
5
votes
0answers
140 views

Measurability of a certain set in Falcolner's Geometry of Fractal Sets

On page 24 of Falcolner's The Geometry of Fractal Sets, Falcolner defines the set $F = \{ x \in E : \mathcal{H}^s(E \cap U) < \alpha$ diam$(U)^s$, for all convex sets $U$ containing $x$ such that ...
1
vote
1answer
102 views

The problem of concentration (Clarification of statement in Evans: Weak Convergence Methods for Nonlinear PDE)

I am working through Evan's book on Weak Convergence Methods for Nonlinear PDE. He assumes that $U$ is an open bounded smooth subset of $\mathbb{R}^n$ and that $1<q<n$. In ($\S$D) concerning ...
3
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1answer
59 views

A question on estimates of surface measures

If $\mathcal{H}^s $ is $s$ dimensional Hausdorff measure on $ \mathbb{R}^n$, is the following inequality true for all $ x \in \mathbb{R}^n,\ R,t > 0 $ ? $$ \mathcal{H}^{n-1}(\partial B(x,t)\cap ...
3
votes
0answers
105 views

Limits to the growth of the volume of a union of spheres

Assume that $x_i$, $i=1,\ldots,m$ are points in $\mathbb{R}^n$, with the maximal distance between any two of them being at most $1$. Define $$ a(r)=\mu\Bigl(\bigcup_{i=1}^m B(x_i,r)\Bigr),$$ where ...
32
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4answers
642 views

To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
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0answers
92 views

doubt in a book proof from 'The Geometry of Fractal Sets'

I am reading the proof of existence of positive finite $H^s$-measure (Theorem 5.4) on page 67-68 of The Geometry of Fractal Sets.I am not quite convinced that $E_k$ are closed set by the construction ...
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0answers
129 views

Change of variables formula for a general measure

In a paper, pp11, I read the equality ...
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0answers
20 views

density of $\mathcal{C}_1$ surface in a point

Let us have an $A \subset \mathbb{R^d}$, that is a $k$-dimensional $\mathcal{C}_1$ surface (obviously $k<d$) and let $a \in A$. Why then is $\Theta^k(A,a)=1?$ Of course $\Theta^k := \lim_{r ...
3
votes
2answers
146 views

$(d-1)$-rectifiability of a boundary of compact convex set

Let us have a compact convex set $A\in \mathbb{R}^d$. Then $\delta A$ should be a $(d-1)$-dimensional rectifiable set. I don't seem to be able to show that it can be covered by a countable union of ...
1
vote
1answer
107 views

Null sets of $\sigma$-algebras generated by functionals

These are some questions that I couldn't answer after a class I'm teaching. I would very much appreciate help. Let $\Omega=[0,1]^N$ and suppose that $f,g\colon \Omega \to [0,1]^N$ are two functions ...
0
votes
1answer
45 views

Approximate a set which is hausdorff measure is infinite.

I have tried to construct a sequence of sets satisfying the following requirement. But I cannot. Could someone help me? Let $A$ be a compact subset of $d$-dimensional $C^1$-manifold embedded in ...
5
votes
1answer
291 views

Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
4
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0answers
174 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...