# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### An exercise about Lebesgue measure.

let $\{E_{n}\}_{n\in\Bbb N}$ be a sequence of measurable subsets such that $E_n \subseteq (0,1)$and let $\limsup_{n\to\infty} m(E_{n})=1$, prove that exists a subsequence {$E_{n_{k}}$} such that ...
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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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### 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 ...
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### 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 ...
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### Measures whose projections are absolutely continuous

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely ...
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### 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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### a measurable subset of the plane with restricted sections

Is there a bounded subset $X\subset \mathbb R^2$ of positive measure such that for almost every line $L\subset \mathbb R^2$ the $1$-dimensional Hausdorff measure of $X\cap L$ is either 0 or 1? If ...
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### 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 ...
### 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 $\mu=Du$ ?