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18 votes
4 answers
3k views

Generalized Stokes' theorem

In the Wikipedia article on Stokes' theorem the following claim is advanced without any references given: The main challenge in a precise statement of Stokes' theorem is in defining the notion of a ...
JaberEdgar's user avatar
9 votes
1 answer
918 views

A Besicovitch-type Covering Theorem

In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
BigbearZzz's user avatar
  • 1,245
7 votes
2 answers
448 views

Uncountable collections of distinct subsets of an interval (existence)

Throughout, $\mu$ is just the Lebesgue measure. Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
Stepan Plyushkin's user avatar
7 votes
1 answer
437 views

Proper homotopy

Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper? I am interested in particular in ...
Onil90's user avatar
  • 823
7 votes
2 answers
243 views

Continuous section of support - Is it possible to map compact sets to measures supported on them?

Let $(X,d)$ be a compact metric space and let $(\mathcal K(X),d_H)$ and $(\mathcal P(X),d_W)$ denote its space of nonempty compact subsets with Hausdorff metric $d_H$, and its space of Borel ...
Christian Bueno's user avatar
7 votes
0 answers
493 views

A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel

I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces: Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
Kaitei's user avatar
  • 99
6 votes
2 answers
483 views

Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure \begin{equation} H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : \...
Greg Zitelli's user avatar
  • 1,124
6 votes
1 answer
212 views

Geometry of complements to compacts of codimension 2

Let $K\subset \mathbb{R}^n$ be a (nonempty) compact of covering dimension $\le n-2$. In particular, $K$ does not separate $\mathbb{R}^n$ (even locally). I will equip $M=\mathbb{R}^n-K$ with the ...
Misha's user avatar
  • 31.2k
6 votes
0 answers
309 views

Have we discovered constructions for natural fractional dimensional spheres?

I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
Sidharth Ghoshal's user avatar
6 votes
0 answers
156 views

Topological properties of the dual of differential forms

Notation: $U \subset R^n$, bounded open set $D^k(U) = \{ \omega : U \to \Lambda^k R^n : \text{compactly supported and infinitely differentiable \}}$ $D_k(U) = D^k(U)'$ is the topological dual space (...
Wreck it Ralph's user avatar
6 votes
1 answer
300 views

Proof of Denjoy-Riesz Theorem and Moore's Generalization?

The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
John Samples's user avatar
4 votes
1 answer
121 views

Nice arrangement of open sets in $\sigma$-algebras

Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Let $O$ be an open ...
ABB's user avatar
  • 4,058
4 votes
1 answer
2k views

Lebesgue measure of boundary of Caccioppoli set

Can anything be said about the measure of the topological boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say ...
Martijn's user avatar
  • 320
4 votes
0 answers
414 views

Topology on the space of Borel measures

Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...
Longyearbyen's user avatar
3 votes
1 answer
376 views

Hausdorff measure of intersection of a ball and a set in $\mathbb {R} ^ n$

Let $A$ a subset of $\mathbb R ^n$, $B=B(x,r) \subset \mathbb {R} ^n$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $\mathbb R ^n$ by $\mathcal H^{n-1}$. Also assume that $\...
XIII's user avatar
  • 747
3 votes
1 answer
119 views

Nice representation of open sets in $\sigma$-algebras in certain circumstances

Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Assume ...
ABB's user avatar
  • 4,058
3 votes
1 answer
123 views

Approximation on separable topological space with size $\mathfrak{c}$

Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function. Q. Is it possible to find a sequence of ...
ABB's user avatar
  • 4,058
3 votes
1 answer
151 views

Do we have uniformization theorems for fractional dimensional spaces?

The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions. I’m interested in how it generalizes for fractional ...
Sidharth Ghoshal's user avatar
3 votes
1 answer
77 views

Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
ABIM's user avatar
  • 5,405
3 votes
0 answers
81 views

Isotopy Classes and Embeddability of Products in $\mathbb{R}^2$

On MSE I asked if the plane contains an uncountable collection of mutually disjoint copies of the Warsaw Circle; it seems to be false, and is probably already known but I'm not sure that anybody has ...
John Samples's user avatar
2 votes
1 answer
300 views

If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected

Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
No-one's user avatar
  • 1,149
2 votes
1 answer
391 views

(n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper http://www.unige.ch/~...
A random mathematician's user avatar
2 votes
1 answer
156 views

Covering of discrete probability measures

Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...
ABIM's user avatar
  • 5,405
2 votes
0 answers
60 views

Is the set $\operatorname{Unif}(0,\frac{1}{n})$ for odd and even $n$ a 2-alternating capacity?

Let $\Omega$ be a complete metrizable space $\mathscr A$ its Borel $\sigma$-algebra and $\mathscr M$ the set of all probability measures on $\Omega.$ Every non-empty subset $\mathscr P \subset \...
Seyhmus Güngören's user avatar
2 votes
0 answers
159 views

What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?

Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $ E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\} $, which ...
XIII's user avatar
  • 747
2 votes
0 answers
212 views

Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define $$L(d,h)=\lim_{\...
Joseph Van Name's user avatar
1 vote
1 answer
172 views

A question about pushforward measures and Peano spaces

Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...
O-Schmo's user avatar
  • 33
1 vote
0 answers
79 views

Conditions for a function to vanish almost nowhere on its support?

Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support. Under which conditions is it true that $f≠0$ (...
fsp-b's user avatar
  • 463
1 vote
0 answers
280 views

Comparing two $\sigma$-algebras

Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$. Q. For which ...
ABB's user avatar
  • 4,058
1 vote
0 answers
67 views

Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$

Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
PepitoPerez's user avatar