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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6
votes
1answer
161 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
254 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
472 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
273 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
225 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 ...
0
votes
0answers
44 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 ...
1
vote
0answers
61 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
146 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
130 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
votes
1answer
69 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 ...
4
votes
0answers
114 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
votes
4answers
765 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!
1
vote
0answers
96 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 ...
1
vote
0answers
153 views

Change of variables formula for a general measure

In a paper, pp11, I read the equality ...
1
vote
0answers
28 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
1answer
246 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
121 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
56 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 ...
6
votes
1answer
374 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
votes
0answers
196 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 ...
2
votes
1answer
119 views

Estimate on the Hausdorff dimension of boundary of balls

I am reading Evans and Gariepy's book on GMT and I have a couple questions: 1) if E is a set of locally finite perimeter, is it true that E is $ \| \partial E\|$- measurable? 2) At a certain point, ...
1
vote
1answer
264 views

Hausdorff dimension of a smooth manifold

I read a book about fractal stating that without proof: every $m$-dimension $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove it?
4
votes
1answer
221 views

The approximating Hausdorff measure is not Borel

This is an exercise taken from Mattila, Geometry of sets and measures in Euclidean space, chapter 4. Exercise. Let $U$ be an open ball in $\mathbb{R}^n$ ($n\ge 2$) such that $d(U)=\delta$ [here ...
3
votes
1answer
82 views

Reference for an integral formula

Good morning, I'm reading a paper of W. Stoll in which the author uses some implicit facts (i.e. he states them without proofs and references) in measure theory. So I would like to ask the following ...
10
votes
1answer
762 views

Inner regularity property of Radon measures in metric spaces

Let us agree to say that $\mu$ is a Radon measure on a metric space $X$ if it is a Borel measure which is finite on compact subsets and is such that: Every measurable subset $A$ is outer regular, ...
4
votes
1answer
134 views

Volume is a Continuous Function

I am working on the following problem: Suppose $C \subset \mathbb{R}^d$ is a compact and non-empty set. Let $C_0 = C$ and let $C_t = \{x \in \mathbb{R}^d : d(x,C) \leq t \}$ for all $t >0$. ...
0
votes
1answer
43 views

The figure of convex set in $S^{n-1}(1)$

If $X$ is a connected convex set in $S^{n-1}(1)$, then what is $\partial X$ ? Recall the following definition. Definition : $X$ is a convex subset of $S^{n-1}(1)$ if any two points in $X$ can be ...
5
votes
0answers
91 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
1
vote
1answer
58 views

Connection between local dimension and Hausdorff measures

Given that the local dimension of a measure $\nu$ at a point $w$ is defined by $$d[\nu](w) = \limsup_{r \to 0} \frac{\log(B(w,r))}{\log r}$$ is it possible to find a measure which has local dimension ...
1
vote
1answer
96 views

Limit Volume of Parallel Sets

Given $F \subset \mathbb R^n$ non empty and $\epsilon > 0$. Let $F_\epsilon$ be $\epsilon$-parallel set of $F$, $$F_\epsilon := \{x \in \mathbb R^n:d(x,F)\le\epsilon\},$$ with $d(x,F):= \inf_{y\in ...
0
votes
1answer
124 views

Is this map on a metric space upper-semi continuous?

Let $(X,d)$ be a metric space and $\mu$ a probability measure. Let $f(x)=\mu(B(x,r))$. Is this map upper semi continuous? I have some other assumptions since this is part of a larger proof but I'm ...
8
votes
1answer
207 views

Hausdorff Dimension of Set of Measure Zero

It's clear that every $A \subset \mathbb R^n $ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n $ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.
20
votes
1answer
475 views

Measure theoretic definition of curl

Is there a good measure theoretic definition of curl? To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet ...
3
votes
0answers
177 views

Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that $\beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\}$. The Fourier ...
2
votes
1answer
218 views

Hausdorff Measure- Lower semi-continuity

By definition , when we are given a set $A \in \mathbb{R}^n$ , $$ H_\delta^{n-1} (\partial A ) = \inf \left\{ \sum_{j=1}^{\infty} \alpha_{n-1}\frac{1}{2^{n-1}} [\operatorname{diam}(U_j)] ^{n-1} ...
2
votes
2answers
61 views

How smooth is the distribution function of a convex polynomial?

Here is a prototype of the problem I have in mind: Let $P:\mathbb{R}^2\rightarrow\mathbb{R}$ be a strictly convex, nonnegative polynomial such that $P(0,0)=0$. Let $\alpha\geq 0$, and consider the ...
26
votes
1answer
785 views

Which sets are removable for holomorphic functions?

Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class ...
2
votes
1answer
153 views

Equality in the Isoperimetric Inequality

Stein and Shakarchi, in their book Real Analysis, the third volume of the Princeton Lectures in Analysis series, give a proof of the isoperimetric inequality for closed rectifiable curves in ...
3
votes
1answer
98 views

Convolution square root of a Frostman measure

A probability measure $\mu$ on $\mathbb{R}^d$ is said to be a Frostman measure if $$\mu(B)\lesssim r(B)^\alpha \ \ \ \ (1)$$ for all open ball $B$, where $r(B)$ denotes the radius and $\alpha>0$. ...
7
votes
0answers
129 views

Higher Order Coarea Formula

I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
1
vote
2answers
107 views

Is critical Haudorff measure a Frostman measure?

Let $K$ be a compact set in $\mathbb{R}^d$ of Hausdorff dimension $\alpha<d$, $H_\alpha(\cdot)$ the $\alpha$-dimensional Hausdorff measure. If $0<H_\alpha(K)<\infty$, is it necessarily true ...
3
votes
0answers
146 views

Integral Geometry Reference Request

I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
1
vote
1answer
400 views

Estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary

Wondering if it is possible to estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary, and that it is know the internal surface area is solid. ...
1
vote
0answers
307 views

Corollary of Lebesgue decomposition theorem and counter-example

Refferring to the Lebesgue decomposition theorem in Lebesgue decomposition theorem and fundamental theorem of calculus there is a corollary when the measure is the Lebesgue measure that states: if ...
1
vote
1answer
222 views

Surface measure for Lipschitz domain

Let $D\subset R^d$ be a bounded Lipschitz domain. Must there exist a bounded function $\Phi$ on $\partial D$ and collections of subsets $(\partial D )^{\epsilon} \subset \partial D $ (indexed by ...
3
votes
0answers
141 views

Radon-Nikodym derivative of the Hausdorff measure transform

Let $H^{m}$ be the $m$-dimensional Hausdorff measure, let $D = \operatorname{diag }(d_1,...,d_k)$ be a nonsingular scale matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; ...
6
votes
1answer
429 views

Is the Hausdorff outer measure regular?

An outer measure $\mu^*$ is said to be regular if for every set $A \subset X$ $$\mu^\ast (A)=\inf\{\mu^*(E) : E\supset A \text{ is } \mu^\ast\text{-measurable} \}$$ To check that an outer ...
0
votes
1answer
143 views

The modulus of curve family

I'm reading book "Geometric function theory and non-linear analysis" and there is one thing that I did not understand. A curve family $\Gamma$ is a collection of curves. An admissible density is ...
0
votes
1answer
113 views

Dimension invariance

Where can I find the Theorem of invariance of the dimension with diffeomorphisms? And about Between Homeomorphism? I also want to know about the Hausdorff dimension invariance what is deeper!
3
votes
0answers
178 views

Co-area formula involving non-integer Hausdorff measure

Is there any co-area formula involving non-integer Hausdorff dimension? Moreover is it sensible to write the following: Let $S$ be a subset in $ \mathbb{R}^n$ with Hausdorff dimension s ...