16
votes
Radon-Nikodym theorem for non-sigma finite measures
The measure spaces for which all of the nice theorems about $\sigma$-finite measures (except those about product measures, which require other generalizations) generalize are called localizable ...
- 3,816
13
votes
Accepted
Sets that project to zero measure on all lines except one
Consider a sequence of rational numbers $s_i$ with the following property: For each rational number $s$ and nonnegative integer $n$, there exists $i$ such that $ s=s_i =s_{i+1} = \dots = s_{i+n}$, and ...
- 148k
11
votes
Accepted
If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Yes, $\mathbb{R}^n\setminus E$ has to be path-connected.
Let $x,y\in\mathbb{R}^n\setminus E$, we will prove that there are many paths going from $x$ to $y$ inside $\mathbb{R}^n\setminus E$. We can ...
- 10.6k
9
votes
Accepted
Bounding an "integral" from below by the Hausdorff measure of the domain
It is true and deserves to be known better (it is morally equivalent to the statement that the Frostman lemma is just an exercise in linear programming duality though there are some pesky details I ...
- 61.9k
9
votes
Accepted
Computing a limit on the unit sphere: Riemann Lebesgue?
The key fact here is the (surprising, initially, but well known) (power) decay of $\widehat{\sigma}(\xi)$. If $u\in C^{\infty}(S)$, we can extend to a function $u_0\in C^{\infty}_0(\mathbb R^d)$, and ...
- 24.2k
9
votes
Accepted
Finiteness of Hausdorff measure of balls
As a counterexample, let $X$ be an infinite-dimensional normed space. For $\varepsilon<r/2$ it follows from Borsuk-Ulam that you need more than $n$ closed sets of diameter $\varepsilon$ to cover ...
- 1,869
8
votes
Accepted
Fubini's theorem for Hausdorff measures
If $s>1$, then clearly $H^s(B_x)=0$ so there is nothing to do. If $s=1$, $H^1$ is just the Lebesgue measure so measurability follows. If $0<s<1$ the situation is a way more complicated, but ...
- 28k
8
votes
Accepted
Hausdorff dimension of the boundary of fibres of Lipschitz maps
Unfortunately, you can always find a Lipschitz map
$$
f:\mathbb{R}^m\to\mathbb{R}^{m-k}
\quad
\text{and}
\quad
y\in\mathbb{R}^{m-k}
$$
such that $\partial f^{-1}(y)$ has positive $m$-dimensional ...
- 28k
8
votes
Sets that project to zero measure on all lines except one
This can be done with an iterated Venetian blind construction.
The idea is you start with a horizontal line segment, replace it with many small rotated parallel line segments, and repeat. In each step,...
- 613
7
votes
Accepted
Is Hausdorff Measure equal to Hausdorff Content on rectifiable (metric) spaces?
In general, no. For example, $X$ may be a countably infinite collection of lines through the origin in $\mathbb{R}^2$. Then $X$ is $1$-rectifiable.
For any ball $B$ centered at the origin, $B\cap X$ ...
- 86
7
votes
Radon-Nikodym theorem for non-sigma finite measures
One can formulate the Radon-Nikodym so that it applies to all measure spaces without any restriction. Let $(X,\Sigma,\nu)$ be a measure space and $\nu$ be a real valued signed measure on $\Sigma$.
...
7
votes
Accepted
Hausdorff measure
Not true for all $X$.
Taking the gauge function $\phi(t) = t^{1/2}/\log|t|$, construct a Cantor set $X$ in $\mathbb R$ using Hausdorff's original method so that $H^\phi(X) = 1$. Then the Hausdorff ...
- 41.1k
6
votes
Accepted
Continuity of Hausdorff measure on level sets
As Leo Moos suggested in the comments, in any dimension $d$, this is a simple consequence of the implicit function theorem.
The implicit function theorem implies that every point $x\in\phi^{-1}(0)$ ...
- 8,992
5
votes
Generalization of area and coarea formula for fractional Hausdorff measures
There are several intersting consequences of this abstract viewpoint about these formulas. I also would have liked to discuss why the coarea inequality is the backbone of the coarea formula, but let ...
- 622
5
votes
Accepted
The product of two Hausdorff measures
As noted in a comment, for Riemannian manifolds, the Hausdorff measures are equal to (up to a constant) the usual volumes. So this works.
The metric case you mention can fail. There are metric ...
- 41.1k
5
votes
Accepted
Hausdorff dimension of Julia set
This argument is from Eremenko-Lyubich survey: Let $f$ be a rational map and $\mu$ an $f$-invariant ergodic measure on $\widehat{\Bbb{C}}$. By Ledrappier-Young entropy formula, $h_\mu(f)$ is the ...
- 3,599
5
votes
If $\mathcal{H}^{n-1}(K)=0$ then $\mathcal{H}^n(K\times \mathbb{R})=0$
Frostman's lemma seems to work for this problem.
Suppose that $H^n(K\times\mathbb{R})>0$. Then $H^n(K\times[0,1])>0$, so there is a measure $\mu$ in $\mathbb{R}^{n+1}$ with $\mu(K\times[0,1])>...
- 10.6k
4
votes
Accepted
Hausdorff dimension and surface measure
Section 3.3.4D in Evans and Gariepy "Measure Theory and Fine Properties of Functions". Chapter 3 of Federer's "Geometric Measure Theory" is also a canonical reference.
- 2,247
4
votes
Average of the sum of dirac measures
I assume that by maximal you mean with respect to inclusion. Then the answer is no. Consider the following counterexample on the real line:
Let $\mathcal{B}_\epsilon := \epsilon\mathbb{Z}$ and $\...
- 2,504
4
votes
Accepted
If $\mathcal{H}^{n-1}(K)=0$ then $\mathcal{H}^n(K\times \mathbb{R})=0$
$\newcommand\de\delta\newcommand\ep\varepsilon\newcommand\R{\mathbb R}\newcommand\N{\mathbb N}$Saúl RM proved the desired result by an application of Frostman's lemma.
Here is an elementary proof:
...
- 128k
4
votes
Accepted
Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure
The claim holds under the much weaker assumption that the exceptional set satisfies $\mathcal H^{n-1}(E)=0$.
Since the limits of $f$ and $\nabla f$ exist everywhere we get two continuous functions $F\...
- 16.4k
4
votes
Accepted
A question about the maximal function
The answer is no. Let $m$ be a very large positive constant. You can find smooth $f$ that equals $m$ on a small ball $B_R(0)$ and still satisfy $\int_{B_6(0)}|f|<\delta$. You can do it with $R$ ...
- 28k
4
votes
Accepted
Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?
Here is a way to do it without the axiom of choice, but it isn't a nice formula either.
Consider a Cantor set $C\subseteq[0,1]$ with Hausdorff dimension $0$. Now consider a countable disjoint union $\...
- 10.6k
3
votes
Hausdorff dimension of Julia set
Here is a fairly elementary argument, which works in great generality (in particular for rational and transcendental entire functions, and indeed suitably modified for meromorphic functions and ...
- 6,548
3
votes
Continuity of Hausdorff measure on level sets
Not a complete answer, but just some ideas that are too long for a comment.
Note: Here we use $\mathcal H^n, \mathcal L^n$ to denote the Hausdorff and Lebesgue measure respectively.
Let $\Omega \...
- 6,223
3
votes
Recovering the length metric from Hausdorff measure
The easiest argument I know (which works for path-metrics on topological manifolds $X$ and in even greater generality) is to consider the induced map $f^2: X^2\to Y^2$. This map also preserves the ...
- 12.3k
3
votes
Accepted
volume of region between two manifolds
If $C_1$ and $C_2$ are the boundary components of an $\epsilon$-neighborhood of $C$, then the volume of the region is $2\epsilon$ times the volume of $C$. This is a special case of the Weyl tube ...
- 6,337
3
votes
Hausdorff approximating measures and Borel sets
There is a difference between $m = n$ and $m < n$. In case $m=n$, all the outher measures $\mathscr H^n_\delta$ are equal to the Lebesgue measure. So each Borel set in $\mathbb R^n$ is $\mathscr H^...
- 920
3
votes
Accepted
Gromov-Hausdorff distance between weighted tree graphs
I would direct you to Theorem 7.3.25 of the book "A Course in Metric Geometry" by Burago-Burago-Ivanov.
Roughly speaking, the Gromov-Hausdorff distance between two compact metric spaces can be ...
- 46
3
votes
Accepted
Why is the Hausdorff measure of this set zero?
The next result answers the question in the negative.
Theorem. There is $\phi:\mathbb{R}^n\supset\Omega\to\mathbb{R}^n$ of class $C^\infty$ such that
$\phi$ is a local diffeomorphism in a ...
- 28k
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