1
vote
1answer
85 views

Two equivalent measures on the real Grassmannians?

Denote by $G(n,k)$ the real Grassmannian, the set of $k$-dimensional subspaces of $\mathbb{R}^n$. It is a topological space, even metrizable (see A metric for Grassmannians), and so it is a ...
1
vote
0answers
74 views

Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...
5
votes
2answers
206 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 : ...
0
votes
0answers
89 views

Convergence of functions of bounded variation

Let $u_n$ be a sequence of smooth functions with $||u_n||_{W^{1,1}}<C<\infty$. By $BV$ compactness, a subsequence of $u_n$ converges to some $u$ in $L^{n/(n-1)}$. Assume $\int_{\Omega} (|\nabla ...
2
votes
1answer
117 views

Function from a compact metric space to the subsets of the naturals

Let $X$ be a compact metric space, and $\mu$ a Borel probability measure. For $S\subset\mathbb{N}$ we denote the upper density with $\overline{D}(S).$ Let $f:X\rightarrow2^{\mathbb{N}}$ be a ...
2
votes
1answer
124 views

Are there any good techniques for calculating Hausdorff measure?

I'm aware that many techniques have been developed for the purpose of calculating Hausdorff dimension (although I'm fairly unfamiliar with them), but my question is whether or not we have any good ...
1
vote
1answer
107 views

Hausdorff measure and projections

Fix $ k \in \mathbb{N} $ and let $ H^k $ be the $k$-dimensional Hausdorff measure on $\ell^\infty $. Also, if $ V $ is a subspace of $ \ell^\infty $, we denote the projection onto $ V $ by $ \pi_V $. ...
2
votes
1answer
155 views

$X$ Polish geodesic implies $(P_2(X), W_2)$ geodesic

If $X,d$ is a complete and separable space then the space of Borel probability measures with finite second moment on $X$ endowed with the Wasserstein distance $W_2$ is geodesic. I am looking for ...
1
vote
0answers
131 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 ...
4
votes
0answers
163 views

decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
0
votes
0answers
161 views

Measurable projection theorem

Hi ; i have this theorem from the book :Set-valued analysis Let $(\Omega,\mathcal{A},\mu)$ be a complete $\sigma$-finite measure space , $X$ a complete separable metric space and ...
5
votes
1answer
212 views

Extension of measures from the ball sigma-algebra to the borel sigma-algebra

Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and $\Sigma_{2}$ the sigma algebra generated by balls (open and closed). If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
14
votes
0answers
689 views

A Kakeya-like problem: must a union of annuli fill the plane?

Let $S$ be a subset of $\mathbb{R}^2$ with the following property. For all $x \in \mathbb{R}^2$ and $\varepsilon \gt 0$, there exists a nontrivial interval $[a,b] \subseteq [1-\varepsilon,1]$, such ...
2
votes
1answer
226 views

common dominating measure for a family of measures

Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that $$\mu_i=f_i \lambda$$ where the f_i are densities ...
7
votes
2answers
508 views

Integration on the space of symmetric matrices

Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi_{A}(x)$ be its ...
3
votes
2answers
622 views

absolute continuity on $R^{n}$

I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$. I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
2
votes
1answer
749 views

Measure of infinite intersection of sets

Given a closed bounded set $X \subset \mathbb{R}^3$ and two curves $\gamma_1$ and $\gamma_2$ in the group of orientation preserving isometries of $\mathbb{R}^3$. Define the sets $X_1$ and $X_2$ as ...
2
votes
2answers
288 views

Existence of rearrangements of functions in $L^p([0,1])$ when given a measure preserving map

Given a funtion $f \in L^p([0,1])$ (take $p=\infty$ if you'd like), and also a measure preserving map $s:[0,1] \to [0,1]$ (meaning $s$ pushes Lebesgue measure forward to itself) I would like to know ...
4
votes
1answer
362 views

Jordan measurability of the level sets

Let $A$ be a compact subset of $R^n$ and $d_S(\bullet, A)$ be the signed distance function of $A$. Namely, $d_S(p,A) = d\left({p,\partial A} \right)$ for p in A, and $d_S(p,A) = -d\left({p,\partial ...
7
votes
2answers
492 views

local behavior of a finite Borel measure

Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I ...
13
votes
3answers
2k views

Is arbitrary union of closed balls in R^{N} Lebesgue measurable?

Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb R^{N}$ Lebesgue measurable? If so, is it a Borel set? @George I still have two questions concerning your sketch of ...
10
votes
2answers
946 views

What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?

I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...