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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34
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4answers
956 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!
29
votes
1answer
1k 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 $\...
29
votes
2answers
637 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 ...
20
votes
1answer
521 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 ...
16
votes
7answers
851 views

Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality?

The notion of having a number $a \in \mathbb{R}_{\geq 0} $ associated to any metric space is described by the definition of a "Hausdorff Dimension". I was wondering if work has been done on spaces ...
14
votes
2answers
218 views

$f : \mathbb{R} \to \mathbb{R}$ (Lipschitz) continuous implies $f(A)$ is Borel for all Borel $A$.

Full question: Let $(\mathbb{R}, \mathfrak{M}, m)$ denote the measure space $\mathbb{R}$ equipped with the Borel $\sigma$-algebra and the Lebesgue measure. Suppose $f : \mathbb{R} \to \mathbb{R}$ is ...
13
votes
2answers
887 views

Relations between various definitions of a Radon measure

The following various definitions of a Radon measure seem to be given for the Borel sigma algebra of different types of topological spaces: general, Hausdorff, locally compact, or locally compact ...
12
votes
1answer
326 views

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
10
votes
1answer
830 views

Uncountable sets of Hausdorff dimension zero

Let $A \subset \mathbb{R}$ be a countable set. It is easy to see that $A$ has Hausdorff dimension $\dim_H(A) = 0$. Do there exist uncountable sets $A \subset \mathbb{R}$ with $\dim_H(A) = 0$?
10
votes
1answer
1k 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, ...
10
votes
0answers
347 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
9
votes
1answer
161 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{...
8
votes
1answer
268 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.
8
votes
0answers
158 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 ...
7
votes
1answer
635 views

Minkowski Content

Could someone provide some intuition behind the $n$-dimensional Minkowski Contentthe $n$-dimensional upper Minkowski Content of $\mathcal{A}$ as $$\mathfrak{M}^{*m} (\mathcal{A}) : = \lim_{\epsilon \...
7
votes
3answers
324 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
7
votes
1answer
190 views

Uniform estimate for the boundary area of the union of closed unit balls

Let $A$ be the union of closed unit balls around countable many points $\{v_i\}\in\mathbb R^n$. If $\{v_i\}$ are chosen from a fixed unit ball, is it true that the boundary of $A$ has area $< c(n)$?...
7
votes
0answers
150 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 ...
6
votes
2answers
1k views

Best textbook for Geometric Measure Theory

I was wondering what is the best textbook for Geometric Measure Theory for self study. I am looking for one that isnt excessively detailed or long either as I found Rana's Introduction to measure ...
6
votes
1answer
596 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 ...
6
votes
2answers
170 views

Methods to define areas

I always thought that areas are defined by integrals, until I read Michael Spivak's Calculus p.289: The desire to define area was the motivation, both in this book and historically, for the ...
6
votes
1answer
568 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!
6
votes
1answer
179 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 $aE=[ax;x\...
6
votes
2answers
391 views

Change of variable within an integral of the Hausdorff measure

Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, $H^{m}$ be a Hausdorff measure. Is it true that $$ \int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx) $$ ...
6
votes
1answer
232 views

A curious problem about Lebesgue measure.

The Problem: Let $(B(x_{m},0.5))_{m}$ be a sequence of disjoint open discs in $\mathbb{R}^{2}$ centered in $x_{m}$ and with radius 0.5. Let $\psi(n)$ be the number of these discs contained in the ...
6
votes
1answer
571 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 ...
6
votes
0answers
128 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 ...
6
votes
0answers
306 views

Ham sandwich theorem for integrable functions?

The classical ham sandwich theorem says that given $n$ measurable sets in $\mathbb{R}^n$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single $(n − ...
5
votes
2answers
183 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 ...
5
votes
2answers
68 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 ...
5
votes
1answer
131 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 ...
5
votes
2answers
172 views

Does interval spacing effect Hausdorff dimension of Cantor set?

Let $C=\bigcap_{j=0}^{2^n}C_j$, $C_0=[0,1]$, and the intervals in the construction of each stage of $C_j$ consists of removing the center 1/3 from the $j-1$ stage intervals. In other words, the ...
5
votes
1answer
249 views

Change of Variables for Hausdorff Measure

(Read bounty text for answering question) Let $H^{m}$ be the $m$-dimensional Hausdorff measure. Let $D$ be a linear transformation matrix. Consider the change of measure formula: $$ \int\limits_{A} ...
5
votes
1answer
265 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 $...
5
votes
2answers
184 views

Borel sets and measurability

Is it always possible to construct a measure $ \mu $ on a Hausdorff space Y such that the $ \mu $-measurable sets are exactly the Borel sets of Y? By Theorem in 2.2.13 of Federer's book this question ...
5
votes
0answers
168 views

Lebesgue density theorem for compact metric spaces.

Let $X$ be a compact metric space (with balls $B_{\varepsilon }(x)$), $\mu $ a Borel probability measure, and $A$ a Borel set with positive probability. Do we have that $\lim_{\varepsilon \...
5
votes
0answers
261 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 ...
5
votes
0answers
154 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 $...
5
votes
0answers
256 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 \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\...
4
votes
2answers
104 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!
4
votes
3answers
388 views

Total variation of (weakly) differentiable functions

the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as $$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert ...
4
votes
1answer
486 views

Lebesgue measure on normal matrices

Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer ...
4
votes
1answer
132 views

Example of a Borel measure, which is not Borel-regular

I have asked a question to find four types of outer measures here, and I could find three of the four examples. We call an outer measure $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ Borel, if ...
4
votes
1answer
438 views

How to understand currents in geometric measure theory?

I find it is hard to catch the current, sometimes it is just the picture as its support set (if I do not miss it). What is the heart idea of the current? What are the benefits to introduce such an odd ...
4
votes
1answer
45 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 ...
4
votes
1answer
415 views

Volume form and Hausdorff measure

Let $M$ be a smooth orientable $(n-1)$-dimensional submanifold in $\mathbb{R}^n$, $dS$ be its volume form and $dH^{n-1}(x)$ be an $(n-1)$-dimensional Hausdorff measure. How to show than that $$ \...
4
votes
1answer
596 views

Calculating the upper Minkowski dimension of the set $\{0,1,\frac{1}{2}, \frac{1}{3}, \ldots \}$

The upper Minkowski dimension of a compact set $A$ in $\mathbb{R}$ is defined as $$ \overline{\dim}_M = \inf \{ \epsilon > 0 : \text{ there is a constant } C(\epsilon) \text{ such that } \mu(A_\...
4
votes
1answer
148 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 $t >0$. ...
4
votes
1answer
105 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 measure-...
4
votes
2answers
125 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 ...