All Questions
17 questions
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 ...
- 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 $...
- 101
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 ...
- 459
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 : \...
- 1,124
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 ...
- 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 ...
- 541
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 $\...
- 747
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 ...
- 4,058
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 ...
- 5,405
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 ...
- 1,149
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:+}(\...
- 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 \...
- 359
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_{\...
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 $f:P\to P$ ...
- 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 $f≠0$ (...
- 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 ...
- 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\...
- 19