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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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 ...
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122 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 ...
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211 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 ...
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163 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 ...
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2answers
181 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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228 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 ...
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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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55 views

Is a bijective projection function measure preserving?

A subspace with dimension strictly less than the dimension of vector space has (Lebesgue) $measure=0$. Let $V$ be a vector space with $dimension=n$. To show that some set $S$ in V is zero-measure, is ...
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1answer
358 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 ...
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32 views

Comass of a differential form

In the wikipedia article on currents https://en.wikipedia.org/wiki/Current_%28mathematics%29 it is written that If $\omega$ is an m-form, then define its comass by $||\omega|| = \sup\{|\langle ...
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97 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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91 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 ...
3
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1answer
158 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 ...
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1answer
45 views

The derivative of a measure

Let $\mu$, $\nu$ be two Radon Measure on $\mathbb{R}^n$. How can I prove that $D_{\mu}{\nu}=\lim_{r \to 0} \frac{\nu(B(x,r)}{\mu(B(x,r))}$ is in $L^1_{loc}(\mathbb{R}^n,\mu)$?
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1answer
56 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
690 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 ...
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1answer
100 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 $\alpha>0$. ...
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1answer
37 views

Integral similar to Lebesgue point theorem

Assume we are in $\mathbb R^N$ and $\Gamma$ is a ($N-1$)-rectifiable set with $\mathcal H^{N-1}(\Gamma)<\infty$ and $\mathcal H^{N-1}(\bar \Gamma\setminus \Gamma)=0$. Let $u\in C_c(\mathbb R^N)$ ...
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1answer
39 views

Fubini type results for Hausdorf dimension?

Suppose that I have a stack of hyperplanes in Euclidean space $\mathbb{R}^n$, let's call each plane $P_{a, x}=\{y\in\mathbb{R}^n\mid \langle y, x\rangle=a\}$ Suppose that a measurable subset $A$ of ...
3
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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
165 views

Is measure $m$ invariant under change of metric?

$\newcommand{\vol}{\operatorname{vol}}$ Let $(M^n, g)$ be a Riemannian manifold. We define $d\vol(g)$ as a canonical volume form and then integrate. It is clear that $d\vol(g)$ depends on the metric ...
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1answer
80 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 ...
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1answer
83 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 ...
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54 views

The projection of density 1 point on a rectifiable set.

This post has also been posted here. Please see the comment on the linked page, useful information! Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal ...
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38 views

When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
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54 views

Sobolev Multiplication theorem for Fibre bundles

Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a manifold admitting a ...
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1answer
56 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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45 views

A covering argument for metric Jacobian

Given a Lipschitz map between Carnot Groups $ f : G_1 \to G_2$, with homogeneous dilations $ \delta^1_s, \delta^2_s$, we have the almost everywhere Pansu derivative $ D_H f(x)(y) = \lim_{s\to 0} ...
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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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84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: 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
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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249 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 ...
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147 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) \; ...
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190 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 ...
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2answers
399 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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2answers
127 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 ...
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1answer
45 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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1answer
309 views

Banach-Tarski Paradox on the middle third Cantor set

In analysis and topology, the middle third Cantor set $C$ is often a very interesting topic of discussion. My question is that is it possible to have some sort of measure preserving bijection $f: ...
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1answer
78 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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1answer
22 views

$\iint |x-y|^{-t} \,d\mu\, d\mu < \infty$ iff $t<1$

Consider the measure space $([0,1], \mathcal{L}([0,1]), \mu)$, where $\mu$ is the restriction of the Lebesgue measure to the closed interval $[0,1]$. I wish to show $\iint |x-y|^{-t} \,d\mu\, d\mu ...
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1answer
50 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
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1answer
529 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 ...
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112 views

Hausdorff measure with non-power test function

At my analysis course some time ago we were told that there is definition of Hausdorff measure through the test functions which are continuous and non-decreasing $h:(0,\infty)\to(0,\infty)$ and ...
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1answer
27 views

Measure of convex hulls

I'm not an expert of this kind of questions, but I can't give a satisfactory answer to the following question. Pick $x_1\dots x_n \in \mathbb{R}^m$. Is there a formula for the measure of the ...
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1answer
15 views

What does it mean “Distance between $k$-planes induced by the identification plane-projection matrix”?

I'm reading some parts of Functions of bounded variation and free discontinuity problems by Ambrosio, Fusco, Pallara. At the very beginning of page 82 there's written "Let $G_k$ be the complete ...
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1answer
44 views

Weak isoperimetric inequality for surfaces in $\mathbb R^3$

The well-known isoperimetric inequality in $\mathbb R^2$ states that for a $\Gamma \subset \mathbb R^2$ a curve (simple, closed, piecewise $C^1$) and $A$ the area of the bounded component of ...
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1answer
59 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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2answers
215 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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1answer
89 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
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1answer
67 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 ...