# 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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### 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 ...
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### doubling property on manifold

I am learning the Geometric Measure Theory, and curious about how to generalize the covering lemma on manifold. However I am stuck at the doubling property: Let $(X,d,\mu)$ be a metric space with ...
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### Borel regular measure on a subset of $\mathbb{R}^3$

Let $f(x,y)$ be a positive, differentiable function defined in the unit disc $D\subset \mathbb{R}^2$ and let $S=\{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2<1, z=f(x,y)\}$ Given $A \subset S$,...
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### Is a bijective projection function measure preserving?

A subspace with dimension strictly less than the dimension of vector space has (Lebesgue) $measure=0$. Let $V$ be a vector space with $dimension=n$. To show that some set $S$ in V is zero-measure, is ...
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### Weak isoperimetric inequality for surfaces in $\mathbb R^3$

The well-known isoperimetric inequality in $\mathbb R^2$ states that for a $\Gamma \subset \mathbb R^2$ a curve (simple, closed, piecewise $C^1$) and $A$ the area of the bounded component of $\Gamma^c$...
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### Theorem on Measurability-Preserving Maps?

How to tell whether you should bother reading this: Consider two well known facts about measurable functions: (i) the supremum of a sequence of measurable functions is measurable, (ii) if $f$ is ...
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### When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
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### Does every bounded Jordan measurable set have porous boundary?

Let $A\subset \mathbb{R}^n$ be a bounded Jordan measurable set. I wonder if its boundary $\partial A$ is necessarily porous. I know that $\partial A$ has Lebesgue measure zero. I also think that one ...
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### Sobolev Multiplication theorem for Fibre bundles

Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a manifold admitting a ...
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### Is an $L^p$ function in an annulus $L^p$ restricted to almost all planes?

Let $n\geq3$ and consider the annulus-like domain $A=B(0,1)\setminus B(0,r)\subset\mathbb R^n$. Take any number $p\in[1,\infty]$. If $f\in L^p(A)$, is it true that $f|_{P\cap A}\in L^p(P\cap A)$ for ...
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I am currently trying to solve some analysis exercises on metric spaces, but I cannot quite tackle on of them. The exercises read as follows: Define the measure $$\mu_x:=\frac{d(\bullet,x)}{\mu(B(x,d(... 1answer 51 views ### The derivative of a measure Let \mu, \nu be two Radon Measure on \mathbb{R}^n. How can I prove that D_{\mu}{\nu}=\lim_{r \to 0} \frac{\nu(B(x,r)}{\mu(B(x,r))} is in L^1_{loc}(\mathbb{R}^n,\mu)? 0answers 43 views ### Uniqueness of measure I'm studying the Lebesgue-Besicovitch differentiation theorem (on the book "Sets of finite perimeter and geometrical variational problem- Maggi") but I'm not able to understand the following: Why the ... 0answers 83 views ### Vector valued measures I can't understand what a vector valued measure is. In particular what does it mean to write \nu (A) with A Borel measurable set? 1answer 42 views ### Property of Radon measure on \mathbb{R}^n Let \mu a Radon measure on \mathbb{R}^n, C a bounded and measurable set and F_i a countable family of closed ball. I also have that \frac{\mu(C)}{\eta(n)}\le \mu (C \cap \bigcup \{ \bar{B} \... 1answer 44 views ### calculus of the measure of a C^1  hypersurface I have to prove that:$$\lim_{r \to 0} \frac {\mathcal H ^{n-1}(M \cap B(x,r))}{\omega_{n-1} r^{n-1}}=1, $$where \mathcal H ^{n-1} is the (n-1)-dimensional Hausdorff measure, M is a C^1 ... 1answer 36 views ### Action of differential on multivectors, what is it called? Let \varphi:X\to Y be a smooth map between finite dimensional vector spaces over \mathbb R. Let us define its differential in a pont a\in X as a (linear) map d\varphi(a):X\to Y acting by ... 1answer 26 views ### Regularity of limit measure and prove that |\mu_h|\stackrel{*}{\rightharpoonup}|\mu| I have some questions. First of all, let \mu_h a sequence of Radon measures and suppose that \mu_h weakly-converge to another measure \mu. Now, this limit measure \mu is still Borel? Is it ... 0answers 47 views ### Does the coarea formula hold for smooth maps with gradient bounded below? The coarea formula for hypersurfaces in \mathbb R^n can be written in two following forms:$$ \int_{\mathbb R^n} g(x) |\nabla u(y)| dx = \int_{\mathbb R} \int_{u^{-1}(t)} g(y) d\mathscr H^{n-1}(y) \...
We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...