# 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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### Hausdorff dimension of a Sierpinski-like triangle

Define the set $A \subset \mathbb R^2$ by proceeding as follows. Let $A_0$ be a closed equilateral triangular region of side 1. $A_1$ are the three equilateral triangular regions of side $\frac 1 3$ ...
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### 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 ...
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### 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 $1\le i\le n, \space 0<c_i<1$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $A_k\in \... 1answer 62 views ### A counterintuitive construction: finely covering a horizontal strip with vertical bands Let$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: ... 0answers 305 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 E_t(u)\|(\Omega)\... 0answers 213 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 ... 0answers 82 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 middle-... 1answer 47 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 ...
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 \$\...