0
votes
0answers
13 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 ...
0
votes
1answer
34 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 ...
2
votes
0answers
21 views

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

Let $\lambda$ be a Radon measure on $\mathbb{R^n}$ and $A \subset \mathbb{R^n}$. Show that $\displaystyle \lim_{r \rightarrow 0} \frac{\lambda(A \cap B(x,r))}{\lambda(B(x,r))}=0 \ \ \ \text{for ...
1
vote
1answer
23 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 ...
1
vote
1answer
50 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
0
votes
1answer
21 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 ...
0
votes
1answer
15 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 ...
0
votes
1answer
19 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 $\sum_i d(E_i)^s \leq \mathcal{H^s_{\delta}}(A)+1$ ...
6
votes
0answers
170 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 ...
0
votes
1answer
32 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, ...
3
votes
2answers
92 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 ...
0
votes
0answers
28 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
2
votes
1answer
51 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 ...
3
votes
1answer
84 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 ...
2
votes
1answer
80 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 ...
1
vote
1answer
29 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 ...
4
votes
2answers
106 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 ...
1
vote
1answer
26 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, ...
0
votes
0answers
30 views

Borel outer measure and Borel measure

I have these two definitions: Given $\ (X,\mathcal{M},\mu)$ measure space, the measure $\ \mu$ is Borel measure if $\ \mathcal{M}=\mathcal{B}(X) $ Given $\ \phi:X\to[-\infty,+\infty]$ be an outer ...
3
votes
0answers
73 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 ...
1
vote
1answer
55 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 ...
0
votes
1answer
60 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 $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
0
votes
0answers
27 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 ...
1
vote
0answers
39 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 [ ...
3
votes
0answers
57 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 ...
1
vote
0answers
36 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 ...
2
votes
2answers
171 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: Given $k<n$ every $k$-dimensional subspace of $\mathbb{R}^{n}$ has ...
2
votes
0answers
36 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 $L^1$ ...
3
votes
4answers
136 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 $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ ...
5
votes
1answer
248 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 ...
6
votes
1answer
156 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 ...
1
vote
1answer
39 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 ...
5
votes
1answer
273 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!
0
votes
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 ...
0
votes
1answer
46 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 ...
0
votes
1answer
38 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
79 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
votes
3answers
106 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
vote
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 $\mu=Du$ ?
2
votes
1answer
106 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
150 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 ...
3
votes
1answer
107 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 ...
1
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 ...
1
vote
1answer
178 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 ...
1
vote
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 ...
1
vote
1answer
52 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
votes
0answers
180 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 ...
1
vote
1answer
50 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} ...
1
vote
1answer
99 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
votes
0answers
102 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 ...