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 ...

learn more… | top users | synonyms

1
vote
1answer
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 ...
0
votes
0answers
30 views
+50

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 ...
0
votes
0answers
16 views

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 ...
1
vote
0answers
11 views

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 ...
1
vote
1answer
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 ...
1
vote
0answers
13 views

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 ...
0
votes
0answers
25 views

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 $2\pi$ ...
0
votes
0answers
18 views

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 $\| \chi_{(a,b)} f\|_\infty = + \infty$?
0
votes
1answer
13 views

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 ...
0
votes
2answers
31 views

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 ...
4
votes
0answers
131 views

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 ...
0
votes
1answer
28 views

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 ...
0
votes
0answers
8 views

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 ...
1
vote
1answer
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 $$ \limsup_{n\to\infty}\int_I\sqrt{1+(f'_n)^2}<\infty $$ ...
1
vote
1answer
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 ...
2
votes
0answers
64 views

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 $BV(\Omega)$. ...
1
vote
1answer
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\} ...
1
vote
0answers
28 views

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 $$ ...
1
vote
2answers
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 ...
1
vote
0answers
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 ...
2
votes
0answers
21 views

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 ...
0
votes
0answers
19 views

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 ...
1
vote
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 ...
2
votes
1answer
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= ...
3
votes
0answers
60 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 ...
4
votes
1answer
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 ...
2
votes
1answer
28 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 ...
2
votes
1answer
17 views

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 $\{x_N=0\}$. ...
0
votes
1answer
39 views

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 ...
1
vote
1answer
33 views

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 ...
-1
votes
1answer
62 views

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 ...
1
vote
0answers
32 views

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 ...
2
votes
1answer
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 ...
1
vote
0answers
8 views

$\mathbf{M}$ 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 \{ ...
0
votes
0answers
12 views

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 ...
3
votes
1answer
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 ...
0
votes
1answer
20 views

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 ...
2
votes
1answer
26 views

Proof step in Rademacher's Theorem

In the proof of Rademacher's theorem, we assume that $f: \Bbb R^n \to \Bbb R$ is a Lipschitz function and $v \in \Bbb R^n$ is a vector with $\Vert v \Vert = 1$. Our aim is to show, that $$ \mathrm ...
1
vote
1answer
48 views

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 ...
1
vote
0answers
37 views

Structure theorem for locally bounded variation functions

I have some doubts reading Measure Theory and Fine Properties of Functions by Evans and Gariepy. In particular, they define the space of locally bounded variation functions $BV_{loc}(U)$ in $U\subset ...
0
votes
1answer
26 views

Total variation of a vector valued measure

If I have understood correctly, a vector valued measure $\mu$ is simply a vector of measures, that is $\mu=(\mu_1,\dots,\mu_n)$, where $\mu_i$ is a possibly signed measure on the measure space ...
0
votes
0answers
27 views

The minimal $C^1$ hypersurface

Let $S_0\subset \mathbb R^N$ be given, where we assume that $S_0$ is a connected $C^1$ hyper surface and $\mathcal H^{N-1}(S_0)<\infty$. Here $S_0$ may not be able to represented as a graph. (up to ...
3
votes
1answer
40 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 $u\in C_c(\mathbb R^N)$ ...
0
votes
0answers
17 views

Can a subset of $N−1$ rectifiable set be partitioned into countably many connected pieces? [duplicate]

This is a follow up question regarding to the problem discussed here. Thanks so much to @Silvia Ghinassi's nice and simple explanation there! Ok, here is the updated question: Let $\Gamma\subset ...
1
vote
1answer
26 views

Can a $N-1$ rectifiable set be partitioned into countably many connected pieces?

Let $\Gamma\subset \mathbb R^N$ be a $N-1$ rectifiable curve such that $\mathcal H^{N-1}(\Gamma)<\infty$. I am wondering that would it be possible to partition it into countably many connection ...
2
votes
1answer
18 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 ...
4
votes
2answers
123 views

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 ...
4
votes
0answers
62 views

Singular points of one-to-one mapping on rectifiable set

Let $E$ be an $s$-rectifiable set in $\mathbb{R}^n$ of positive $s$-dimensional Hausdorff measure $H^s(E)>0$. The original question I have is: Can there exist a one-to-one Lipschitz function ...
0
votes
2answers
38 views

Hausdorff dimension of homeomorphic compact metric spaces

Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff ...
2
votes
1answer
40 views

Find parametrization for a possible “ruled” surface in $\mathbb R^4$

Let us endow $\mathbb R^4$ with a group law $\cdot$ such that the dilations $\delta_\lambda:(\mathbb R^4,\cdot)\to (\mathbb R^4,\cdot), (x_1,x_2,x_3,x_4)\mapsto (\lambda x_1,\lambda x_2,\lambda^2 ...