All Questions
Tagged with hausdorff-dimension measure-theory
24 questions
2
votes
0
answers
68
views
An algebraic condition possibly related with the Hausdorff measure on $\mathbb{R}$
This is my first time to ask a question here. Please tell me if I can improve it.
I would like to introduce the following definition inspired from a measure theory exercise.
Definition. A subset $K$ ...
4
votes
1
answer
552
views
Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?
Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, and $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ be the Hausdorff measure in its ...
- 63
15
votes
4
answers
1k
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Steinhaus theorem and Hausdorff dimension
Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem ...
- 28k
2
votes
2
answers
850
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Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha\in[0,2]$
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff ...
- 63
5
votes
0
answers
160
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Naïve definition of a measure on a fractal
This question was previously posted on MSE.
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use ...
- 333
8
votes
1
answer
866
views
Fubini's theorem for Hausdorff measures
$B\subset \mathbb{R}^2$ is a Borel set. Define the slices $B_x:= \{y \in \mathbb{R}: (x,y) \in B \}$.
If $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, presentations of Fubini's theorem often ...
- 675
4
votes
1
answer
903
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Hausdorff dimension and surface measure
Could someone please indicate me some reference that contains the proof of the following theorem?
Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$.
Theorem: ...
- 649
3
votes
1
answer
151
views
For $\mathcal{L}^1$-a.e. $t\in\mathbf R$, is $n-1$ the Hausdorff dimension of level sets of a locally Lipschitz function $f:\mathbf R^n\to\mathbf R$?
Let $f:\mathbf R^n\to\mathbf R$ be a locally Lipchitz function. Denote $\mathrm H^n$ the $n$-dimensional Hausdorff measure. We know that for any $\mathrm H^n$-measurable subset $A\subset\mathbf R^n$, ...
8
votes
1
answer
213
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How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?
Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask:
Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
- 12.7k
3
votes
1
answer
230
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Hausdorff dimension and $W^{1,1}$ functions
What can be said about the Hausdorff dimension of the image of a set by a $W^{1,1}$ map?
In other words,
what is the relationship between
$\mathrm{dim}_H f(A)$ and $\mathrm{dim}_H A$, where $f \in ...
- 839
23
votes
3
answers
1k
views
Existence of subset with given Hausdorff dimension
Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension.
For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
- 1,045
13
votes
1
answer
577
views
Are Hausdorff measures on the real line Haar measures for some locally compact topology?
For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...
- 32.5k
4
votes
0
answers
119
views
A quantity that distinguishes finer than Hausdorff dimension
Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have ...
- 1,648
1
vote
0
answers
114
views
density of fractal measures
Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...
- 11
9
votes
1
answer
637
views
Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 1,035
2
votes
1
answer
388
views
Construction of null sets with prescribed Hausdorff dimension and generalizations
Given $h:\mathbb{R}_0^+ \to \mathbb{R}_0^+$ increasing and right continuous, the outer measure $\mathcal{H}^h$ in $\mathbb{R}^d$ that assigns to every $E\subset\mathbb{R}^d$ the measure $$\mathcal{H}^...
- 193
4
votes
1
answer
969
views
Usable Change-of-Variables Formula for Hausdorff Measure
Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
9
votes
1
answer
1k
views
When is Hausdorff measure a Frostman measure?
Let $(X,d)$ be a metric space and let $\mathcal{H}^s$ be the $s$-dimensional Hausdorff measure on $X$.
For a measure $\mu$ on $X$, we say that $\mu$ is a Frostman measure (sometimes referred as ...
- 398
6
votes
2
answers
1k
views
Hausdorff dimension of a Cantor-like set
Suppose $K$ is a subset of $[0,1]$ with the following property: for almost $x,y \in K$, we have
$$\frac{x+y}{2} \not\in K.$$
(Here, "almost in $K$" means "in $K$ except for a countable subset").
...
- 373
7
votes
3
answers
679
views
How can dimension depend on the point?
Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
- 8,086
6
votes
1
answer
670
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Calculate Hausdorff measure with Frostman measures
Fix a metrix space $(X,d)$ and consider the Hausdorff (outer) measures $\mathcal{H}^s$ on $X$.
A Frostman measure on $X$ is a finite Borel measure $\mu$ such that there exists $C,t,r_0>0$ with $\...
- 9,650
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 : \...
- 1,134
3
votes
1
answer
581
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 ...
- 173
8
votes
2
answers
388
views
Isometrically-invariant measures and dilation of the Cantor set
Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. ...
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