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
24 views

Minkowski dimension behaviour of sets with positive finite Hausdorff measure.

It is (rather) well known that the set \begin{equation*} E=\{k^{-1},k\in\mathbb{N}^{*}\} \end{equation*} has box-dimension $1/2$ and Hausdorff dimension $0$. However $H^{0}(E)=|E|=+\infty$. Is it ...
2
votes
2answers
99 views

Evaluating the limit for the Minkowski content of $F_{\alpha}=\{0,1,\frac{1}{2^{\alpha}},\frac{1}{3^{\alpha}},\frac{1}{4^{\alpha}},\dots\}$.

The Minkowski content is defined as $\displaystyle M_{\beta}(A)=\lim_{\delta \rightarrow 0} \frac{\mu(A_{\delta})}{(2\delta)^{1-\beta}}$ where $0 < \beta < 1$, $A \subset \mathbb{R}$, and ...
0
votes
1answer
17 views

Easy argument in Lemma of Corea formula

I don't understand a presumably easy argument in my textbook. Let $L: \mathbb R^n \to \mathbb R^m$ be a linear map, $n \geq m$, $A \subset \mathbb R^n$ $\lambda^n$-measurable. We assume that $\dim ...
0
votes
1answer
27 views

Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$

I am reading Lebesgue Integration on Euclidean Space by Frank Jones. My question is specifically regarding Chapter 4, Section C titled "The Lebesgue Function Associated with a Cantor Set". The author ...
0
votes
2answers
28 views

Steiner symmetrization of Lebesgue measurable set

I'm reading a proof in "Evans / Gariepy: Measure theory and fine properties of functions" of the following statemant: Theorem: Let $A \subset \mathbb R^n$ be $\lambda^n$-measurable, $a \in \mathbb ...
2
votes
2answers
70 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 ...
0
votes
1answer
23 views

Understanding integral equation with locally integrable function

I'm reading a proof in a book and don't understand a crucial step. So we have $f \in L^1_{\text{loc}}(\mathbb R^n)$ and $\phi \in C_c(\mathbb R^n)$. Now $\mu$ is a Radon measure, such that $$ \mu(A) ...
1
vote
1answer
42 views

Prove carpet has positive Hausdorff measure in its dimension

Given $D\subset\{0,1,2,\dots n-1\}\times\{0,\dots,m-1\}$, let $$K(D)=\{\sum_{k=1}^\infty(a_kn^{-k},b_km^{-k}):(a_k,b_k)\in D\forall k\}.$$ Show that if $D$ has uniform horizontal fibers (i.e. the ...
-1
votes
1answer
98 views

Proving ineqalities for the similarity dimension

a. Let $K$ be the attractor of the IFS $\{f_1,\dots f_n\}$ which satisfies SSC (i.e $f_i(K)\cap f_j(K)=\emptyset\forall i\neq j$) where for all $i, c_i$ such that $ 1\le i\le n, \space ...
1
vote
1answer
51 views

Bounding dimension of IFS

Given the IFS $\{\frac x {2+x},\frac 2 {2+x}\}$ ($0\le x \le 1$) with attractor K prove that $0.53<\dim_HK<0.8$ I thought using the results from my last question by saying ...
2
votes
1answer
44 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 [ ...
0
votes
0answers
25 views

Sard's Theorem with different Measures

From what I can tell Sard's theorem is formulated in terms of the Lebesgue measure. Is there a form of Sard's theorem for more general measures (in particular, those which are not absolutely ...
0
votes
1answer
26 views

Upper Minkowski content and finite Hausdorff measure

Does someone know an example of a set $E$ with positive finite $s$-Hausdorff measure, Minkowski dimension $s$, and infinite $s$-dimensional upper Minkowski content ? The $s$-dimensional upper ...
2
votes
0answers
55 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial ...
1
vote
0answers
24 views

Define Radon measure as an integral

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon measure and $f \in L^1_{loc}(\mathbb R^n, \mu)$, $f \geq 0$ on $\mathbb R^n$. Now we define an outer measure $\nu: \mathbb R^n \to ...
0
votes
0answers
18 views

Proving that $\dim_M{K_{\lambda,\gamma}}=\dim_HK_{\lambda,\gamma}$

Define $K_{\lambda,\gamma}$ to be the attractor of the IFS $$\bigg\{\bigg(\matrix {\lambda&0\\0&\gamma}\bigg)\bigg(\array{x\\y}\bigg),\bigg(\matrix ...
4
votes
0answers
34 views

Co-Area formula in Riemannian geometry

I wonder if the following holds true: Let $z:[0,1]\times B^{n-1}_r(0)\to(M^n,g), (t,p)\mapsto z(t,p)$ be a diffeomorphism a.e. onto its image with respect to the Lebesgue measure on $A:=[0,1]\times ...
1
vote
1answer
28 views

Differentiation theorem for Radon measures

I have trouble to understand a detail in the proof of the following Theorem: Theorem: Let $\nu, \mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be outer Radon measures, such that $\nu \ll \mu$. Then ...
1
vote
2answers
39 views

A property of Radon Measures

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon-measure. This means that every Borel-set $B \subset \mathbb R^n$ is $\mu$-measurable, $\mu$ is Borel-regular, i.e. for every set $A ...
1
vote
0answers
24 views

Coarea formula for fractional dimension

The coarea formula states that any locally Lipschitz function (e.g. a $C^1$-function) $F:\mathbb{R}^N\to\mathbb{R}^n$ with $N\geq n$ satisfies $$\int_A JF(x) \mathrm{d}\mathcal{H}^N = ...
2
votes
1answer
20 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 ...
1
vote
0answers
93 views

How we can explain this by drawing a suitable diagram?

When I go through the proof of Riesz-Markov Representation Theorem I could encounter a step of defining $A_k=K\cap f^{-1}((y_k,y_{k+1}))$ $U_k=V\cap f^{-1}((y_k,y_{k+1}))$ And the $A_k\in ...
14
votes
7answers
597 views

Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality?

The notion of having a number $a \in \mathbb{R}_{\geq 0} $ associated to any metric space is described by the definition of a "Hausdorff Dimension". I was wondering if work has been done on spaces ...
0
votes
0answers
21 views

Poincaré inequality by capacity estimate.

Let $(X,d,\mu)$ be a complete metric space with a doubling measure $\mu$. For any ball $\mu(B)< \infty$ and $\mu(A)=\sup\{\mu(K);K\subset A, K \text{ compact}\}$. For any two closed disjoint ...
0
votes
1answer
53 views

A counterintuitive construction: finely covering a horizontal strip with vertical bands

Let $Q:=[0,1]^2$ and suppose we have a horizontal strip $S=\{\alpha\le y\le\beta\}\cap Q$ inside it (with $0\le\alpha<\beta\le 1$). We want to cover $S$ in an accurate way with vertical bands: ...
0
votes
0answers
44 views

The level set of Lipschitz functions

Suppose $u$: $R^N\to R$ is lipschitz, then do we have a.e. level set of $u$ has Lipschitz boundary? Is this anything to do with Sard theorem? Sard theorem states that a.e. Level set of smooth ...
7
votes
3answers
275 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
25
votes
1answer
744 views

Which sets are removable for holomorphic functions?

Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class ...
-2
votes
1answer
40 views

Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
6
votes
1answer
341 views

Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
1
vote
0answers
55 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
0
votes
0answers
33 views

Proving that plane - cantor - set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$. Denote the orthogonal projection of the set from the ...
1
vote
1answer
29 views

Fractal Dimension of $C_{\frac{1}{3}}\times[0,1]$

I wonder what is the dimension of the fractal set given by the product of the unit interval $[0,1]$ by the thirds-cantor-set ($C_\frac{1}{3}=\bigcap_n C_n$ where $C_0=[0,1],C_1=[0,\frac 1 3]\cup[\frac ...
3
votes
1answer
33 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 ...
1
vote
2answers
34 views

A conjecture about lines and points in the plane

Let $ l_1, l_2, \ldots$ be an infinite sequence of lines in the plane, and let $(a_1, b_1), (a_2, b_2), \ldots$ be an infinite sequence of pairs of points such that $a_i, b_i \in l_i$ and $a_i \neq ...
1
vote
1answer
46 views

Is this c the same as that c?

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
1
vote
0answers
35 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same ...
1
vote
1answer
34 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
1
vote
2answers
48 views

Estimating the missing points of a 3D point cloud

Consider a cloud of N points (forming a smooth 3D object), in which n points are missing. Also, consider that there is no prior knowledge about the original shape of the point cloud. The only ...
0
votes
0answers
19 views

Existence of Lebesgue measure in parametric function space

I am thinking about this question but I can not solve. The question is: Can I define Lebesgue measure in the space of parametric functions and if the answer is yes what is that Lebesgue measure? Could ...
1
vote
1answer
28 views

A question about the Integral geometry and geometric probability.

In the book: Integral Geometry and Geometric Probability, (p16-17), the author proved that the measure of randomly throwing three points P1, P2, and P3 on the plane such that the circumdisk and the ...
0
votes
0answers
35 views

Showing that $\mathcal{H}^s$ is Borel regular (assuming we know already know that $\mathcal{H}^s$ is measure)

I am trying to show that $\mathcal{H}^s$ (s-dimensional Hausdorff measure) is Borel regular. I am using the defintion $\mathcal{H}^s_{\delta}(F)=inf \Bigg\{ \sum_{i=1}^{\infty}|V_i|^s : \{V_i\} \text{ ...
1
vote
1answer
28 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 ...
1
vote
1answer
39 views

Proving that a mass distribution has positive Lebesgue measure

I am confused in this proof about how we obtain $\int f(u) \, d\mu(u) = \int f(u)g(u) \, d\mu(u)$ and how Plancherels theorem has been applied in $(6.6)$. Furthermore, I cannot understand how if ...
0
votes
1answer
31 views

Hausdorff measure and Ahlfors regular set

Let $E \subset \mathbb{R}^n$ be a closed set, not reduced to a point, and let $\mu$ be a (positive Borel) measure supported on $E$. If there is a constant $C \ge 1$ such that $C^{-1} r^d ≤ μ(E \cap ...
1
vote
0answers
87 views

Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
0
votes
0answers
11 views

Ahlfors regular

Let F be a closed set of $R^n$. Let $\mu$ be a borel regular measure on $R^n$. Let there be a constant C>0 such that for all $x\in F$, all $r\in (0,1]$ and all $\lambda \geq 1$ with $\lambda r \leq ...
0
votes
1answer
31 views

Find $\dim_H \operatorname{proj}_\theta E$ where is the circular Cantor set in the complex plane

$\newcommand{\proj}{\operatorname{proj}}$ $\dim_H(\cdot)$ is the Hausdorff dimension of a set. Let us denote the line through the origin in $\mathbb{R}^2$ which makes an angle of $\theta$ with the ...
0
votes
1answer
47 views

Showing that a precursor to the packing measure (on $\mathbb{R}^n$) is not a measure

I am trying to prove the highlighted sentence. What countable dense sets should I consider? and how am I trying to prove this is not a measure? I am using the usual definition of a measure (and do ...
1
vote
1answer
38 views

Use of Banach-like Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified.