# 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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### To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
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### Hausdorff Dimension of Set of Measure Zero

It's clear that every $A \subset \mathbb R^n$ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n$ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.
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### Higher Order Coarea Formula

I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
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This is an exercise taken from Mattila, Geometry of sets and measures in Euclidean space, chapter 4. Exercise. Let $U$ be an open ball in $\mathbb{R}^n$ ($n\ge 2$) such that $d(U)=\delta$ [here $... 2answers 184 views ### Borel sets and measurability Is it always possible to construct a measure$ \mu $on a Hausdorff space Y such that the$ \mu $-measurable sets are exactly the Borel sets of Y? By Theorem in 2.2.13 of Federer's book this question ... 0answers 168 views ### Lebesgue density theorem for compact metric spaces. Let$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 \...
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 ...