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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371 views
Which sets are removable for holomorphic functions?
[Note: I received a version of this question via email and decided to answer it on MSE, where it might be useful to others.]
Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some ...
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votes
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417 views
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!
4
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1answer
235 views
Lebesgue measure on normal matrices
Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer ...
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0answers
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Limits to the growth of the volume of a union of spheres
Assume that $x_i$, $i=1,\ldots,m$ are points in $\mathbb{R}^n$, with the maximal distance between any two of them being at most $1$. Define
$$ a(r)=\mu\Bigl(\bigcup_{i=1}^m B(x_i,r)\Bigr),$$
where ...
2
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1answer
245 views
Banach-Tarski Paradox on the middle third Cantor set
In analysis and topology, the middle third Cantor set $C$ is often a very interesting topic of discussion. My question is that is it possible to have some sort of measure preserving bijection $f: ...
