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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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 ...
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20 views

Graph (or manifold) Lipschitz satisfy the sphere (ball) condition?

Consider $\varphi: U\subset \mathbb{R}^{n-1}\to \mathbb{R}$ a Lipschitz function and $\Omega=Graph(\varphi)$, i.e., $$\Omega=\{x=(x_1,...,x_n)\in U\times\mathbb{R};x_n=\varphi(x_1,...,x_{n-1})\}.$$ ...
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20 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 ...
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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 ...
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16 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, ...
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20 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 ...
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56 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 ...
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20 views

Show that $d\log f$ is a 1-current on a 1-dimensional complex manifold

I am having trouble with this problem (and it might be because I have the formulation slightly off). I need to show that $d\log f$ is a 1-current on a 1-dimensional complex manifold $M$. This means ...
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1answer
39 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 ...
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36 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 $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
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20 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 ...
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31 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 [ ...
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1answer
33 views

Local Approximation $W^{1,p}$ functions by smooth functions.

I am working through "Measure Theory and Fine Properties of Functions" by Evans and Gariepy, and Chapter 4 section 2 focuses on approximating $W^{1,p}$ functions. In particular, I am confused by ...
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73 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
52 views

Is Hausdorff measure continuous with respect to Hausdorff distance?

Say I have a sequence of surfaces, $S_n$ in $\mathbb{R}^k$ with $d$-dimensional Hausdorff measure $m_d(S_n)$. Say $S_n$ converges in the Hausdorff distance sense to a surface $S$ with $d$-dimensional ...
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28 views

Problem with definition of pushforward density

Let $X$, $Y$ be smooth manifolds and let $\pi \colon X \to Y$ be a submersion. Then for every $y \in Y$ the set $W_y = \pi^{-1}(y)$ is a submanifold in $X$. Let $\mu \in \Gamma(|\Lambda| X)$ be a ...
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48 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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55 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 ...
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32 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 ...
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125 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 ...
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27 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 $L^1$ ...
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106 views

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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125 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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143 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 ...
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1answer
133 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 ...
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1answer
30 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 ...
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1answer
203 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!
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1answer
68 views

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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22 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 ...
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38 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 ...
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1answer
27 views

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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48 views

Linearising angle to chord length over a reasonable domain of < pi; or, how to make measuring a rock with a protractor easy

Imagine that I have a protractor and compass, and wish to use it to measure the distance between two points (potentially in three dimensional space, such as on a rock). However also being a forgetful ...
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1answer
44 views

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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41 views

Isometric trivialization of tangent bundle of Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold. $x\in M$. We know that there is a neighborhood $U$ of $x$ we can have a isometric trivialization $\pi$ of $TM$, Given by the orthonormal frame on $U$. Formally, ...
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1answer
36 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 ...
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76 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 ...
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57 views

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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1answer
149 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
53 views

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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3answers
82 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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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 $\mu=Du$ ?
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89 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 ...
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116 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 ...
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1answer
27 views

Uniform Rectifiability

What is the definition of uniform rectifiability as used in the context of analytic capacity of compact sets in $\mathbb{C}$? The precise context is this paper by Mattila, Mernikov and Verdera.
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1answer
86 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
34 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 ...
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1answer
120 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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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 ...
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1answer
41 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 ...
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155 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 ...