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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42 views

Measure defined in an atypical way

I was reading a paper when I found this ($\partial \Omega$ refers to the boundary of $\Omega$ and $\nabla$ to the gradient operator,$\nabla f = (\partial_{i}f)_{i} $ ). Let $\Omega \subset ...
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53 views

Two Radon measures and mutual singularity

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$. Show that $\mu$ and $\lambda$ are mutually singular iff $D(\mu,\lambda,x)=\infty$ for $\mu$ almost all $x \in \mathbb{R^n}$. I have looked ...
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60 views

Measure derivatives and the chain rule

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$ such that $\mu << \lambda$. Prove that $\displaystyle \int D(\mu,\lambda,x)^2 d\lambda x= \int D(\mu,\lambda,x)d\mu x$. Is it ...
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67 views

Density and $\lambda$ - measurability of a Radon measure

Question: Let $\lambda$ be a Radon measure on $\mathbb{R}^n$ and $A \subset \mathbb{R}^n$. Show that $$ \lim_{r \rightarrow 0} \frac{\lambda(A \cap B(x,r))}{\lambda(B(x,r))}=0 \ \ \ \text{for ...
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37 views

Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.

Let $\Omega\subset\mathbb{R}^N$ be a open, unbounded and connected set ($N\ge 2$). Let $m$ and $\mathcal{H}^{N-1}$ denote respectively, Lebesgue and $(N-1)$-Hausdorff measures. Suppose that ...
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52 views

Showing that the upper packing dimension is the packing dimension

I cannot see how the first inclusion in this proof works. $P$ is the maximum number of disjoint $B(\epsilon/2)$ with centres in $A$ and the following will help. Moreover I cannot see how it ...
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42 views

Equalities for the Upper and Lower Minkowski dimension definition

In a Geometric Measure Theory textbook the following was written: I cannot see how any of these equalities hold and dont believe they are obvious. If they are relatively obvious could someone ...
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60 views

Properties of the Hausdorff measure

This comes from a book on geometric measure theory in a chapter introducing the Hausdorff measure $\mathcal{H^t}$. I cannot see in this proof how $\sum_i d(E_i)^s \leq \mathcal{H^s_{\delta}}(A)+1$ ...
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93 views

Symmetrization Methods

I was wondering if I could get a list of the symmetrization methods out there i.e. methods that rigidly transform a set A into it's equimeasure ball $A^{*}$. Here are some: a) Steiner Symmetrization ...
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312 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 ...
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69 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
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59 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
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85 views

What is the minimum Number of closed balls covering a boundary as radius $r\to 0$?

Here is the problem: Given compact set $A\subset \mathbb{R}^{d}$, cover $\partial A$ by closed balls $\{B_{i,\varepsilon}\}_{i=1}^{n}$ , with minimum overlap. Can we express n as a factor of ...
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66 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
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879 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 ...
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43 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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45 views

Measure on Locally Compact, Separable metric space

Simon (Geometric Measure Theory) says: If $X$ is a locally compact, separable metric space and $\mu(K) < \infty$ for all $K$ compact, then $X=\cup_{i=1}^\infty U_i$ where $U_i$ are open and ...
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111 views

Constructing an example related to Vitali's covering theorem

An exercise in Fremlin's measure theory vol 2 asks to construct a family $\mathcal I$ of open intervals in the real line such that every point of $\mathbb{R}$ belongs to arbitrarily small intervals of ...
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53 views

Difference of two subharmonic functions and signed measures

One of the reasons subharmonic functions are interesting is that if you take their laplacian, you get a measure (and conversely any finite Radon measure with compact support can be obtained in this ...
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1answer
95 views

The walk of a knife

"A knife is slowly moved parallel to itself over the top of a cake. At each instant the knife is poised so that it could cut a unique slice of the cake. As time goes by the potential slice increases ...
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1answer
204 views

How to correctly calculate the fractal dimension of a finite set of points?

The box-counting dimension is defined by: $\lim\limits_{\epsilon \to 0} \dfrac{N(\epsilon)}{1/ \epsilon}$ What works well if you are solving algebraically or if you can recursively generate more ...
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1answer
145 views

Lipschitz manifold and semi-algebraic is Lipschitz graph?

It is known that there are Lipschitz manifolds that are not Lipschitz graphs and $C^1$ manifold is locally $C^1$ graph. If $M\subset \mathbb{R}^m$ is a Lipschitz manifold (with the outer distance) ...
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169 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 $< ...
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116 views

Measurability of upper and lower derivatives of Radon measures

Let $\mu$ and $\nu$ be Radon measures in $\mathbb R^N$. Define their upper and lower derivatives by $$ \overline{D}_\nu\mu(x):=\limsup_{r\to0}\frac{\mu(B_r(x))}{\nu(B_r(x))},\qquad ...
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106 views

Graph (or manifold) Lipschitz satisfy the sphere (ball) condition?

Consider $\varphi: U\subset \mathbb{R}^{n-1}\to \mathbb{R}$ a Lipschitz function and $\Omega=Graph(\varphi)$, i.e., $$\Omega=\{x=(x_1,...,x_n)\in U\times\mathbb{R};x_n=\varphi(x_1,...,x_{n-1})\}.$$ ...
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1answer
162 views

Projection of measurable sets

If $ X $ and $ Y $ are metric spaces, $ f: X \rightarrow Y $ is lipschitzian and $ H^k $ is the Haussdorf measure, it is easy to check that $ f(A) $ is $H^k $-measurable whenever $ A $ is $H^k ...
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169 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 ...
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1answer
39 views

Hausdorff dimension and accumulation points on a smooth curve

I am wondering about the following, possibly naive, question. Suppose I have a smooth curve, which intersects the horizontal axis in a manner that leads to an accumulation point. More precisely, ...
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142 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 ...
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1answer
152 views

A Borel measure defines semi-continuous function?

Let $X$ be a metric space with outer measure $\mu$, which is assumed to be a Borel measure, i.e., all Borel sets are measurable. For a fixed subset $A\subset X$ (not necessarily measurable, but you ...
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123 views

Indicator function of a level set

Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$. Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
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39 views

Why Steiner Symmetrization makes a measurable set to a measurable one?

I find the Steiner Symmetrization is very useful in proving that the Hausdorff measure coincide with Lebesgue in the Euclidean space. However, I never saw anybody mention that the Steiner ...
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1answer
77 views

What is the density of the SRB measure conditioned to unstable manifolds?

I have a question regarding the SRB measure. As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ ...
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1answer
47 views

Local Approximation $W^{1,p}$ functions by smooth functions.

I am working through "Measure Theory and Fine Properties of Functions" by Evans and Gariepy, and Chapter 4 section 2 focuses on approximating $W^{1,p}$ functions. In particular, I am confused by ...
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84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book MaƱe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
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1answer
182 views

Is Hausdorff measure continuous with respect to Hausdorff distance?

Say I have a sequence of surfaces, $S_n$ in $\mathbb{R}^k$ with $d$-dimensional Hausdorff measure $m_d(S_n)$. Say $S_n$ converges in the Hausdorff distance sense to a surface $S$ with $d$-dimensional ...
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77 views

Problem with definition of pushforward density

Let $X$, $Y$ be smooth manifolds and let $\pi \colon X \to Y$ be a submersion. Then for every $y \in Y$ the set $W_y = \pi^{-1}(y)$ is a submanifold in $X$. Let $\mu \in \Gamma(|\Lambda| X)$ be a ...
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1answer
89 views

Upper Bound on Surface Area of Algebraic Surfaces

I have a family of multi-variate polynomials, with bounded degree and a bound on the norm of the coefficients, say for example, all multi variate polynomials of the form: $\{ a_1 xy + a_2x + a_3xyz: ...
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66 views

BV Function times characterstic function still BV?

I am wondering: given a function $u \in BV(\Theta)$ where $\Theta$ is an open subset in $\mathbb{R}^n$ and a Borel subset $B \in \mathcal{B(\Theta)},$ is the function $w \colon= u \chi_B$ still in ...
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43 views

Aproximating a function on SO(3)

By $SO(3)$ I mean rotation matrices. Let $\cal{L}=\{f:[0,l]\to \rm{SO}(3)\} \cap L^1([0,l];\mathbb{R}^{3\times3})$. How to approximate funkctions from $\cal{L}$ with functions from ...
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2answers
394 views

Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: If $k<n$, then every $k-$dimensional subspace of $\mathbb{R}^{n}$ has ...
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66 views

$L^1$ approximate continuity points of epigraph of a continuous function

Consider a continuous function $f\colon \mathbb R\to \mathbb R$. Let $F(x,y):=\mathrm{sign\,}(y-f(x))$. Is it true that for a.e. $x\in \mathbb R$ the point $(x,f(x))$ is not a point of $L^1$ ...
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2k views

A formula for Perspective measurement

If I have a photograph of a rectangular object whose image recedes with perspective, is it possible to work out the scale of measurements along that object? For example: I have a photograph of a ...
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4answers
165 views

Prove that $\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$

Suppose that the measurable sets $A_1,A_2,...$ are "almost disjoint" in the sense that $\mu(A_i\cap A_j) = 0$ if $i\neq j$. Prove that $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ ...
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484 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 ...
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1answer
213 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 ...
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1answer
62 views

Assumptions for definition of Radon measure

My reference is L. Simon's Lectures on Geometric Measure Theory. He defines a measure on a set $X$ as a countably subadditive function $\mu:2^X\to[0,\infty]$ with $\mu(\emptyset)=0.$ When $X$ is a ...
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1answer
495 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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183 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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1answer
107 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 ...