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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calculus of the measure of a $C^1 $ hypersurface

I have to prove that: $$\lim_{r \to 0} \frac {\mathcal H ^{n-1}(M \cap B(x,r))}{\omega_{n-1} r^{n-1}}=1, $$ where $\mathcal H ^{n-1}$ is the ($n-1$)-dimensional Hausdorff measure, $M$ is a $C^1$ ...
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26 views

Action of differential on multivectors, what is it called?

Let $\varphi:X\to Y$ be a smooth map between finite dimensional vector spaces over $\mathbb R$. Let us define its differential in a pont $a\in X$ as a (linear) map $d\varphi(a):X\to Y$ acting by ...
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18 views

Regularity of limit measure and prove that $|\mu_h|\stackrel{*}{\rightharpoonup}|\mu|$

I have some questions. First of all, let $\mu_h$ a sequence of Radon measures and suppose that $\mu_h$ weakly-converge to another measure $\mu$. Now, this limit measure $\mu$ is still Borel? Is it ...
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15 views

Does the coarea formula hold for smooth maps with gradient bounded below?

The coarea formula for hypersurfaces in $\mathbb R^n$ can be written in two following forms: $$ \int_{\mathbb R^n} g(x) |\nabla u(y)| dx = \int_{\mathbb R} \int_{u^{-1}(t)} g(y) d\mathscr H^{n-1}(y) ...
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18 views

Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...
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9 views

Prove that $\int_{\mathbb R^n} u_\epsilon \, divT=\int_F div(T*\rho_\epsilon). $

Let $F\subseteq \mathbb R^n$ be a set and let $\chi_F$ its indicator function, regularized with functions $\rho_\epsilon\in C^\infty_c(\mathbb R^n)$ such that $u_\epsilon:=\chi_F* \rho_\epsilon\to 0$ ...
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23 views

Any relationship between Hausdorff measures

Let $ S_1= ( [0,1], d_1 ) $ and $ S_2 = ( [0,1], d_2 ) $ be two metric spaces, where $ d_1 = |x - y|$ and $d_2 = (1/2^i) $ where binary expansion of x and y matches upto $ i^{th} $ coordinate. Let $ ...
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37 views

Mean Curvature Flow

Recently I am reading the mean curvature flow from the lecture notes of Carlo Mantegazza where I found that Under mean curvature flow given by$$\begin{cases}{\partial\over \partial ...
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1answer
33 views

Regularity of $\phi$ in order that $\int g_h \phi \,dx \to \phi(0)$

Define the sequence of functions $(g_h)_h$ where $$g_h(x):= h\, \chi_{[0,1/h]}(x)$$ and the sequence of measures $$(\mu_h(dx))_h:= g_h(x)\,dx.$$ We want to show that $\mu_h ...
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41 views

Show limit exists of quotient of measures

This is a Theorem from Mattila's Book Geometry of sets and measures in Euclidean spaces: Let $\mu$ and $\nu$ be uniformly distributed Borel regular measures on a separable metric space $X$. There ...
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24 views

On a proof of essential uniqueness of uniformly distributed measures

In the book 'Geometry of sets and measures in Euclidean space' by P. Mattila, theorem 3.4 states that Let $\mu$ and $\nu$ be uniformly distributed Borel regular measures on a seperable metric space ...
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8 views

Properties of finite difference approximation to perimeter $|E\Delta (E+\epsilon)|/\epsilon$.

Let $E\subset\mathbb{R}$ be a measurable set. If $E$ is a finite union of disjoint intervals, then $E$ is said to have finite perimeter and its perimeter $P(E)$ can be expressed as the limit $$ P(E) ...
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26 views

Existence of a measure where diameter of balls can be estimated by their measure

In my last post I asked the following question: Given a metric space $X$, let $B_r(x):=\{y\in X\mid d(x,y)\leq r\}$ be the closed balls. Fix a real number $p>0$. My question now is: Is there a ...
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17 views

Existence of a measure defined on balls

Given a metric space $X$, let $B_r(x):=\{y\in X\mid d(x,y)\leq r\}$ be the closed balls. Fix a real number $p>0$. My question now is: Is there a Borel-measure $\mu$ on $X$ such that \begin{align*} ...
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40 views

Prove Property of Doubling Measure on $\mathbb{R}$

Problem. Let $\mu$ be a fixed finite measure on $\mathbb{R}$. $\mu$ is said to be doubling if there exists a constant $C>0$ such that for any two adjacent intervals $I=[x-h,x]$ and ...
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86 views

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to ...
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25 views

Inequality with supremum of divergence of vector fields

I would like to write this inequality: $$\sup\left\{ \int_{A \cup (\mathbb{R}^d \setminus F)} \operatorname{div} T \space \rm d m^d : T \in C_0^1(\mathbb{R}^d,\mathbb{R}^d), |T(x)| \le 1 \right\} \le ...
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20 views

Blow-up of derivative of BV function at the jump set

"Motivation" Let $u\in BV(\mathbb{R}^n)$ be a function of bounded variation, and let $x\in J_u$ be a point in its jump set. For $\mathcal{H}^{n-1}$-a.e. such $x$, we can define the unit normal $\nu$ ...
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35 views

Proving a Certain Planar Measure Is Zero on Horizontal Lines

Question: Suppose $\mu$ is a measure on $\mathbb{R}^{2}$ with respect to which all open squares are measurable. Suppose $\mu$ has the following property: there exists a constant $\alpha\geq 1$ ...
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36 views

Hausdorff measure of a subset of $\mathbb R^3$

Let $f \in L^1_{\text{loc}}(\mathbb R^3)$. We define $A \subset \mathbb R^3$ as $$ A := \left\{ x \in \mathbb R^3 \, : \, \limsup_{r \to 0} \frac 1 r \int_{\mathbb B(x,r)} \vert f(y) \vert \, \mathrm ...
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37 views

Hausdorff dimension of a Sierpinski-like triangle

Define the set $A \subset \mathbb R^2$ by proceeding as follows. Let $A_0$ be a closed equilateral triangular region of side 1. $A_1$ are the three equilateral triangular regions of side $\frac 1 3$ ...
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22 views

A property on exterior measures on a metric space

Let $(X,d)$ be a metric space and $\mu:2^X\to[0,\infty]$ be an exterior measure on $X$, $\mathcal{M}:=\{E\subset X:\forall S\subset X\ \ \ \mu(S)=\mu(S\cap E)+\mu(S-E)\}$. Suppose ...
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148 views

Is there a well-defined notion of measure zero on topological manifolds?

We extend the concept of measure zero on manifolds by local parameterization. but in this definition we have to check if it is true for every parametrization. In Guillemin's Differential Topology this ...
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23 views

Weak Compactness Therem for $L^p$

I have a problem to understand a point in the Weak Compactness Theorem from the book of Evans/Gariepy. So we have $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ a Radon measure with $\mu \ll ...
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21 views

Reconstructing a measure from its (absolutely continuous) marginals

Let's denote by $C$ the space of continuous functions $[0,T] \rightarrow \mathbb{R}^n$ for some fixed $T>0$ and assume we have a probability measure $Q$ on the space $C$. Consider the evaluation ...
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3answers
59 views

Can I cover a square with many line segments?

Not sure If I've chosen the tags correctly. Anyway, is it possible to obtain a unit square with enough line segments oriented vertically, placed next to each other? We know that a unit square has ...
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85 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
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564 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 ...
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1answer
29 views

Differences in defining the packing (outer) measure

The definition of a packing measure in Falconer's Fractal geometry is given by I am assuming that $\mathcal{P}^s(F)$ as defined in 3.24 is an outer measure (this is not stated in the book). Now ...
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27 views

Vaguely relative compact subsets of Radon measures on $\sigma$-compact Polish spaces

Let $ (X, d) $ denote a Polish space and let $M(X)$ denote the space consisting of all finite signed Radon measures on $(X,d)$. We say that a sequence $ \{ \mu_k \}_{k\geq 1} \subset M(X) $ converges ...
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Wasserstein space of order 2.

I have a question about Wasserstein space. I am just wondering if the following statement is true Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with $\bar{\Omega}$ being a Polish space ...
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63 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 ...
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30 views

Slicing rectifiable sets with maps into the sphere

Notation and Context (Might not be helpful) Let $S\subset\mathbb{R}^n$ be a countably $\mathcal{H}^{M}$-rectifiable set. If $f:\mathbb{R}^n\to\mathbb{R}^k$ is a Lipschitz function, then we can find ...
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Prove the set is Jordan- measurable and find the appropriate Jordan measure (volume) of the set V.

Prove the set is Jordan- measurable and find the appropriate Jordan measure (volume) of the set V. $$V= \{ (x,y,z)|x^2+y^2\geq 1, x^2+y^2\leq2x, x^2+y^2+z^2 \geq 4, x+y+z\leq 8\} $$ I'm not sure if ...
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1answer
40 views

Image of $A\subset \mathbb R^d$ under a Lipschitz function is $H^d$ measurable.

I really need an help with the following exercise. Suppose that $A\subseteq \mathbb R^d$ is Lebesgue measurable. Let $f\colon A \to \mathbb R^k $ be a Lipschitz function. Show that $f(A)$ is $H^d$ ...
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30 views

Hausdorff measure of $f(A)$ where $f$ is a Holder continuous function.

Let $f\colon \mathbb R^d\to \mathbb R^k$ be a $\beta-$ Holder continuous function ($\beta \in (0,1)$) and $A\subset \mathbb R^d$. As for a Lipschitz function $g$ it holds that $H^s(g(A))\leq Lip(g)^s ...
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1answer
15 views

$H^s(A)=0 \iff H^s_\infty(A)=0$

I have the following exercise: Show that $H^s(A)=0 \iff H^s_\infty(A)=0$ for $A\subset \mathbb R^d$. Here $H^s(A)$ is he Hausdorff measure of the set $A$, so $H^s(A):=\lim_{\delta\to 0^+} ...
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Does the coarea formula hold for delta-function?

Let $\Omega \subset \mathbb R^n$ be an open bounded domain, $u \colon \Omega \to \mathbb R$ be a Lipshitz function and suppose that $\nabla u (x) \neq 0$ for $x \in \Omega$. The coarea formula tells ...
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67 views

Bounded bessel functions in an s-set projection proof

The following is an extract from Falconer's Geometry of Fractal Sets about the proof of: "...Using the definition of a Bessel function $J_0=\frac{1}{2\pi}\int^{2\pi}_0 \cos(u \cos \theta) ...
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19 views

Easy argument in Lemma of Corea formula

I don't understand a presumably easy argument in my textbook. Let $L: \mathbb R^n \to \mathbb R^m$ be a linear map, $n \geq m$, $A \subset \mathbb R^n$ $\lambda^n$-measurable. We assume that $\dim ...
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1answer
51 views

Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$

I am reading Lebesgue Integration on Euclidean Space by Frank Jones. My question is specifically regarding Chapter 4, Section C titled "The Lebesgue Function Associated with a Cantor Set". The author ...
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42 views

Steiner symmetrization of Lebesgue measurable set

I'm reading a proof in "Evans / Gariepy: Measure theory and fine properties of functions" of the following statemant: Theorem: Let $A \subset \mathbb R^n$ be $\lambda^n$-measurable, $a \in \mathbb ...
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2answers
124 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of H’s and T’s. Let N denote the number of tosses until you see “TH” for the first time. For example, for the sequence HTTTTHHTHT, we needed N = 6 ...
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32 views

Understanding integral equation with locally integrable function

I'm reading a proof in a book and don't understand a crucial step. So we have $f \in L^1_{\text{loc}}(\mathbb R^n)$ and $\phi \in C_c(\mathbb R^n)$. Now $\mu$ is a Radon measure, such that $$ \mu(A) ...
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1answer
58 views

Bounding dimension of IFS

Given the IFS $\{\frac x {2+x},\frac 2 {2+x}\}$ ($0\le x \le 1$) with attractor K prove that $0.53<\dim_HK<0.8$ I thought using the results from my last question by saying ...
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1answer
110 views

Proving ineqalities for the similarity dimension

a. Let $K$ be the attractor of the IFS $\{f_1,\dots f_n\}$ which satisfies SSC (i.e $f_i(K)\cap f_j(K)=\emptyset\forall i\neq j$) where for all $i, c_i$ such that $ 1\le i\le n, \space ...
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44 views

Upper Minkowski content and finite Hausdorff measure

Does someone know an example of a set $E$ with positive finite $s$-Hausdorff measure, Minkowski dimension $s$, and infinite $s$-dimensional upper Minkowski content ? The $s$-dimensional upper ...
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39 views

Minkowski dimension behaviour of sets with positive finite Hausdorff measure.

It is (rather) well known that the set \begin{equation*} E=\{k^{-1},k\in\mathbb{N}^{*}\} \end{equation*} has box-dimension $1/2$ and Hausdorff dimension $0$. However $H^{0}(E)=|E|=+\infty$. Is it ...
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36 views

Sard's Theorem with different Measures

From what I can tell Sard's theorem is formulated in terms of the Lebesgue measure. Is there a form of Sard's theorem for more general measures (in particular, those which are not absolutely ...
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36 views

Define Radon measure as an integral

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon measure and $f \in L^1_{loc}(\mathbb R^n, \mu)$, $f \geq 0$ on $\mathbb R^n$. Now we define an outer measure $\nu: \mathbb R^n \to ...