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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Can you determine the length of a curve by the lengths of its projections onto planes?

If $\Gamma \subset \mathbb R^n$ is 1-rectifiable, then its Hausdorff measure is equal to its integralgeometric measure. That is, $$\mathcal H^1(\Gamma) = \int\limits_{G(1,\mathbb R^n)} \int\limits_K ...
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40 views

Are every measure on $\mathbb{R}^{n}$ borel and/or regular?

I saw the above question in an exam paper and I am not sure how to even start. The question is true for the case of Lebesgue measure but I am not sure for arbitrary measures. I have tried looking at ...
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110 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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78 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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246 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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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 ...
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350 views

Radon-Nikodým derivative with respect to the Lebesgue/Hausdorff measure is always defined

Question 1. Is it possible to say that the Radon-Nikodým derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to the Lebesgue measure is always defined but may be a generalized ...
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Lower semi continuity for the norm of the speed

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz ...
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Bounding below Hausdorff measure of conected set

I'm trying to prove that for every connected set $E\subset\mathbb{R}$, $H^1(E)$, the Hausdorff measure is bounded below by $\text{diam}(E)$. In the answer there, it was suggested to use a Lipschitz ...
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11 views

Can we bound a Brenier map between uniform distributions with the Hausdorff distance between their supports?

Let $A,B$ be compact subsets of $\mathbb{R}^n$. Let $\mu_A$ (resp. $\mu_B$) be uniform probability measures over $A$ (resp. $B$). Then as a consequence of Brenier's theorem there is a one-to-one ...
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Identification of the normal cycle to a closed curve

I'm studying the normal cycles in Morvan's book "Generalized curvatures"; He says that the normal cycle of a domain D bounded by a smooth curve is the current ( i.e the linear continuous functional ...
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Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ ...
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Existence of a locally essentially unbounded integrable function

Does there exist an integrable function $f\colon [0,1]\to \mathbb{R}_+$ such that for every $0\leq a < b\leq 1$ we have $\| \chi_{(a,b)} f\|_\infty = + \infty$?
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233 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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Non-measurable subset of a null set.

I am reading measure theory,and I am searching an example in which a measurable null set have a non-measurable subset because this is the reason that,s why we are studying about complete ...
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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 ...
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Characterizing isotropic measures

A Borel measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$. This means that in ...
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definition of $f$ being measurable

I am having troubles understanding the definition of a function being measurable. I have that for a measure space $(\Omega, \mathcal{F}, \mu)$ a function $f: \Omega \to \mathbb{R}$ is measurable if ...
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Uniform random samples inside bounded region

In an $n$ dimensional space I have a region bounded by pairs of hyperplanes: \begin{equation} b_j \le \sum_{i=1}^n a_{ij} x_i \le c_j, \quad\forall j=1,\ldots,m. \end{equation} We can include in those ...
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21 views

Use uniform convergence to control the derivative of function

Let $f_n$: $I:=[-1,1]\to \mathbb R$ be given, and we also assume that $f_n\in C^\infty(I)$. Assume $f_n\to 0$ uniformly in $I$, and we know $$ \limsup_{n\to\infty}\int_I\sqrt{1+(f'_n)^2}<\infty $$ ...
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55 views

Generalizing Green's Theorem

Let $\phi:[0,1] \rightarrow \mathbb{R}^2$, with $\phi(t)=(x(t),y(t))$, a function satisfying the following assumptions: (i) $x(t)$ and $y(t)$ are absolutely continuous; (ii) $\phi(0)=\phi(1)$, the ...
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Pointwise convergence of a sequence of approximate limits of BV functions.

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
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32 views

The level set of Sobolev function

Let $u\in W^{1,2}(\Omega)$ where $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary. Moreover, we have $0\leq u\leq 1$. Let set $\Gamma$ be defined as $$ \Gamma:=\{x\in\Omega,\,\, u(x)=0\} ...
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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 ...
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approximate $C^1$ function by Holder function

I am trying to prove the following Lemma: Suppose $u$: $\mathcal R^N\to \mathbb R$ is $C^1$. Then for each $\epsilon>0$, there exists a $C^{1,\alpha}$ function $\tilde f$ such that $$ ...
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42 views

Example of a compact non-$G_{\delta}$ set?

For a homework assignment, I recently solved the following problem in Folland's real analysis text. Folland's Question: If $X$ is a Locally Compact Hausdorff space, and $K \subset X$ is a compact ...
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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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23 views

Topologies on Spaces of k-rectifiable sets, and $C^k$ convergence

I've been thinking about topologies on the spaces of k-rectifiable sets (Hausdorff metric topology, varifold topology, etc) in $\mathbb{R}^n$, and I'm wondering if there are hypotheses under which ...
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Estimating Self-Convolution of Surface Measure on Sphere

Let $n\geq 2$, and let $\sigma$ denote the standard surface measure on the $n$-sphere $S^{n-1}$, normalized to have total measure $1$. According to the exercise at the end of Terence Tao's blog post ...
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n-1 dimensionnal Hausdorff measure and codimension 1 measure

I've been told that on a n-dimensionnal Riemannian manifold, the Hausdorff measure of dimension n-1 and the codimension 1 measure $v_{-1}$ (defined below) are mutually absolutely continuous. I've ...
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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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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 ...
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A $\mathcal{C}^{1}$ differentiable domain and Hausdorff dimension estimates

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary. Suppose now, that there exists $R>0$ such that the set $\partial E \cap ...
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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 ...
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43 views

Push forward of the Lebesgue measure is the Haar measure of the Carnot group

I have the following problem. I have a Carnot group $(\mathbb G,*)$ which is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ is stratified as $\mathfrak g= ...
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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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What is the definition of Compact rectifiable set?

In dimension $\mathbb R^N$, we say a set $S$ is $N-1$-rectifiable if there exists a countably many $C^1$ hyper surface $\Gamma_i$ so that $$ \mathcal H^{N-1}(S_u\setminus \bigcup\Gamma_i)=0 $$ Now I ...
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1answer
30 views

Generalizing the Cantor Set to the $n$-dimensional plane

I am interested in how to describe an $n$-dimensional cantor set. I think that it may be a good idea to develop the Cantor Set on the two-dimensional plane at first, but I am having issues figuring ...
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107 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 ...
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Finding the similarity dimension of a variation of the Cantor Set.

If we take the Cantor set and instead of removing the interval $[1/3, 2/3]$, we remove the open interval $[x,1-x]$, with $0<x<1/2$, will the similarity dimension change? What I think is that we ...
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32 views

Inclusions for certain types of measures

Let's use the following definitions: Definition. A measure $\mu: \mathcal P(X) \to [0, \infty]$ is what some authors call a outer measure, i.e. (1) $\mu(\emptyset) = 0$. (2) If $A, A_k ...
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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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The $\mathcal H^{N-1}$ measure of the projection of a curve is always less then itself

Let $\Gamma \subset \mathbb R^N$ be a $\mathcal H^{N-1}$-rectifiable set such that $\mathcal H^{N-1}(\Gamma)<\infty$. Let $P(\Gamma)$ be the projection of $\Gamma$ onto the hyperplane $\{x_N=0\}$. ...
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Use of a Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified. The books gives no justification.
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Hausdorff dimension calculation related to Jarnik's theorem

Let $$F=\{x \in R:||qx||\le2q^{1-\alpha}\log q \text{ for infinitely many } q \in \mathbb{R}\}$$ Show for $\alpha>2$, $\dim_H F\le 2/\alpha$. Jarnik's theorem (By Falconer) says: Suppose ...
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fractal curve and fractal set

Would it be correct to say that all fractal curves are fractal sets, but not all fractal sets are fractal curves? If that is correct, what would be an example of a fractal set that is not a fractal ...
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Hausdorff Measure under linear maps

We have a linear map $L : \mathbb R^n \rightarrow \mathbb R^m $ ($m\geq n$) I have two questions: How does one prove that $\mathcal H^n (L(B(x,r)))=\mathcal L^n (O^* \circ L(B(x,r)))$? (We have ...
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$\mathbf{M}$ is $\mathbf{F}$ lower semicontinuous on $\mathscr{D}_m$

I am working on a problem in Frank Morgan's Geometric Measure Theory book. What I have done so far: $T_i \to T$ under the real flat norm: $\forall \epsilon>0, \exists N$ such that $\min \{ ...
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Cone criterion for partially Lipschitz map

I'm currently reading an article in which there are some techniques that I would like to understand, I hope that someone can help me figuring out the following. Suppose we have a function $G\colon ...
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38 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 ...