The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential and Riemannian geometry, as well as calculus of variations. Statements ...
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46 views
Alternative rigorous definition of a surface integral
Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
2
votes
1answer
60 views
About measure theoretic interior and boundary
Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery.
I just want to clarify whether these definitions of measure theoretic interior
and boundary are correct. Given ...
4
votes
2answers
79 views
Is the validity of measuring area by approximation an assumption of calculus?
The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
1
vote
0answers
27 views
Approximate tangent planes and densities
I'm studying the proof of the following theorem (Simon 'Lectures on Geometric measure theory' Theorem 11.6):
Theorem Let $ M \subset R^{n+k} $ be $ H^n $-measurable (where $ H^n $ is the Haussdorf ...
5
votes
1answer
89 views
Homogenous measure on the positive real halfline
Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where ...
2
votes
1answer
22 views
The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$
I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
0
votes
1answer
40 views
Riemannian measure and Hausdorff measure in a general Riemannian Manifold
Let $ M $ be a Riemannian manifold and let $ \mu $ be its Riemannian measure. This is the measure obtained by Riesz reprersentation theorem such that for every continuous function with compact support ...
0
votes
1answer
37 views
Radon-Nikodým derivative with respect to the Lebesgue/Hausdorff measure is always defined
Question 1. Is it possible to say that the Radon-Nikodým derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to the Lebesgue measure is always defined but may be a generalized ...
2
votes
1answer
49 views
A question about the proof of Rademacher theorem
I'm referring to the proof of Rademacher theorem due to C.B.Morrey (i'm reading it on Simon: 'Lectures on geometric measure theory').\
The proof can be summarized in the following steps:\
1)For every ...
0
votes
0answers
32 views
A question about Lebesgue measure 3
Let $ L^k $ be the k-dimensional lebesgue measure. Let $ A \subset R^n $ be a Borel set. Suppose we have proved that $ L^1(A \cap l )=0 $ for each line $ l $ parallel to some line passing throught the ...
1
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0answers
34 views
A question about Hausdorff measure
Let $ \mathscr{H}^m $ be the m-dimensional Hausdorff measure in $ R^n $. Let $ m<k $ and $ A \subset R^n $. If $ \mathscr{H}^m(A) < \infty $ then $ \mathscr{H}^k (A) =0 $. How can i prove this ...
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0answers
52 views
Almost every restriction is absolutely continuous
I'd like to prove the following: let $B_r(x)$ be the open disk of centre $x$ and radius $r$ contained in $\mathbb{R}^2$, and let $f \in H^1(B_1(0))$ ($H^1 = W^{1,2}$). Fix $\rho < 1$.
Then, for ...
5
votes
0answers
122 views
Measurability of a certain set in Falcolner's Geometry of Fractal Sets
On page 24 of Falcolner's The Geometry of Fractal Sets, Falcolner defines the set $F = \{ x \in E : \mathcal{H}^s(E \cap U) < \alpha$ diam$(U)^s$, for all convex sets $U$ containing $x$ such that ...
1
vote
1answer
44 views
The problem of concentration (Clarification of statement in Evans: Weak Convergence Methods for Nonlinear PDE)
I am working through Evan's book on Weak Convergence Methods for Nonlinear PDE. He assumes that $U$ is an open bounded smooth subset of $\mathbb{R}^n$ and that $1<q<n$. In ($\S$D) concerning ...
3
votes
1answer
44 views
A question on estimates of surface measures
If $\mathcal{H}^s $ is $s$ dimensional Hausdorff measure on $ \mathbb{R}^n$, is the following inequality true for all $ x \in \mathbb{R}^n,\ R,t > 0 $ ?
$$ \mathcal{H}^{n-1}(\partial B(x,t)\cap ...
2
votes
0answers
58 views
Limits to the growth of the volume of a union of spheres
Assume that $x_i$, $i=1,\ldots,m$ are points in $\mathbb{R}^n$, with the maximal distance between any two of them being at most $1$. Define
$$ a(r)=\mu\Bigl(\bigcup_{i=1}^m B(x_i,r)\Bigr),$$
where ...
26
votes
4answers
418 views
To show that the set point distant by 1 of a compact set has Lebesgue measure $0$
Could any one tell me how to solve this one?
Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$
Show that $A$ has Lebesgue measure $0$.
Thank you!
0
votes
0answers
70 views
doubt in a book proof from 'The Geometry of Fractal Sets'
I am reading the proof of existence of positive finite $H^s$-measure (Theorem 5.4) on page 67-68 of The Geometry of Fractal Sets.I am not quite convinced that $E_k$ are closed set by the construction ...
1
vote
0answers
77 views
1
vote
0answers
17 views
density of $\mathcal{C}_1$ surface in a point
Let us have an $A \subset \mathbb{R^d}$, that is a $k$-dimensional $\mathcal{C}_1$ surface (obviously $k<d$) and let $a \in A$. Why then is $\Theta^k(A,a)=1?$
Of course $\Theta^k := \lim_{r ...
3
votes
2answers
52 views
$(d-1)$-rectifiability of a boundary of compact convex set
Let us have a compact convex set $A\in \mathbb{R}^d$. Then $\delta A$ should be a $(d-1)$-dimensional rectifiable set.
I don't seem to be able to show that it can be covered by a countable union of ...
1
vote
1answer
56 views
Null sets of $\sigma$-algebras generated by functionals
These are some questions that I couldn't answer after a class I'm teaching. I would very much appreciate help.
Let $\Omega=[0,1]^N$ and suppose that $f,g\colon \Omega \to [0,1]^N$ are two functions ...
0
votes
1answer
28 views
Approximate a set which is hausdorff measure is infinite.
I have tried to construct a sequence of sets satisfying the following requirement. But I cannot. Could someone help me?
Let $A$ be a compact subset of $d$-dimensional $C^1$-manifold embedded in ...
4
votes
0answers
83 views
minimizing the total variation of BV function with given trace on the boundary of the domain
In 1-2 papers, Sternberg, Williams, and Ziemer proved the following result: if $\Omega$ is a bounded connected open set in $\mathbb{R}^n$ whose boundary is smooth and has positive mean curvature ...
5
votes
1answer
118 views
Hausdorff Dimension of Arbitrary Julia Set
I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$.
I know this question is known for a number of ...
4
votes
0answers
102 views
Three properties of the Lebesgue measure on $\mathbb{R}^n$
I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$.
It is a non-negative ...
2
votes
1answer
43 views
Estimate on the Hausdorff dimension of boundary of balls
I am reading Evans and Gariepy's book on GMT and I have a couple questions:
1) if E is a set of locally finite perimeter, is it true that E is $ \| \partial E\|$- measurable?
2) At a certain point, ...
1
vote
1answer
108 views
Hausdorff dimension of a smooth manifold
I read a book about fractal stating that without proof: every $m$-dimension $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove it?
4
votes
1answer
107 views
The approximating Hausdorff measure is not Borel
This is an exercise taken from Mattila, Geometry of sets and measures in Euclidean space, chapter 4.
Exercise. Let $U$ be an open ball in $\mathbb{R}^n$ ($n\ge 2$) such that $d(U)=\delta$ [here ...
3
votes
1answer
66 views
Reference for an integral formula
Good morning,
I'm reading a paper of W. Stoll in which the author uses some implicit facts (i.e. he states them without proofs and references) in measure theory. So I would like to ask the following ...
8
votes
1answer
276 views
Inner regularity property of Radon measures in metric spaces
Let us agree to say that $\mu$ is a Radon measure on a metric space $X$ if it is a Borel measure which is finite on compact subsets and is such that:
Every measurable subset $A$ is outer regular, ...
4
votes
1answer
96 views
Volume is a Continuous Function
I am working on the following problem:
Suppose $C \subset \mathbb{R}^d$ is a compact and non-empty set. Let $C_0 = C$ and let $C_t = \{x \in \mathbb{R}^d : d(x,C) \leq t \}$ for all $t >0$. ...
1
vote
1answer
81 views
Hausdorff measure of $n$-dimensional cube
This is a home work problem that I am stuck on even though it feels like it should be easy:
Show that the n-dimensional Hausdorff measure of an $n$-dimensional cube is positive and finite. I can ...
0
votes
1answer
36 views
The figure of convex set in $S^{n-1}(1)$
If $X$ is a connected convex set in $S^{n-1}(1)$, then what is $\partial X$ ?
Recall the following definition.
Definition : $X$ is a convex subset of $S^{n-1}(1)$ if any two points in $X$ can be ...
4
votes
0answers
53 views
How difficult is it to impose a differential structure on a fractal?
Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
1
vote
1answer
35 views
Connection between local dimension and Hausdorff measures
Given that the local dimension of a measure $\nu$ at a point $w$ is defined by
$$d[\nu](w) = \limsup_{r \to 0} \frac{\log(B(w,r))}{\log r}$$
is it possible to find a measure which has local dimension ...
0
votes
0answers
63 views
How Wiener Measure on $F(C([0,T]))$ is a Gaussian Measure
I'm looking for some simple proofs for the fact that on $(C[0,T],F(C([0,T])),P_{*})$ where $F$ represents Borel Sigma algebra , $P_{*}$ the Wiener Measure , then how to proove that $P_{*}$ measure is ...
1
vote
1answer
54 views
Limit Volume of Parallel Sets
Given $F \subset \mathbb R^n$ non empty and $\epsilon > 0$. Let $F_\epsilon$ be $\epsilon$-parallel set of $F$, $$F_\epsilon := \{x \in \mathbb R^n:d(x,F)\le\epsilon\},$$ with $d(x,F):= \inf_{y\in ...
0
votes
1answer
60 views
Is this map on a metric space upper-semi continuous?
Let $(X,d)$ be a metric space and $\mu$ a probability measure. Let $f(x)=\mu(B(x,r))$. Is this map upper semi continuous?
I have some other assumptions since this is part of a larger proof but I'm ...
8
votes
1answer
121 views
Hausdorff Dimension of Set of Measure Zero
It's clear that every $A \subset \mathbb R^n $ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n $ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.
13
votes
0answers
181 views
Measure theoretic definition of curl
Is there a good measure theoretic definition of curl?
To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet ...
3
votes
0answers
84 views
Fourier dimension of a measure restricted to an open set
Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that $\beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\}$. The Fourier ...
2
votes
1answer
111 views
Hausdorff Measure- Lower semi-continuity
By definition , when we are given a set $A \in \mathbb{R}^n$ ,
$$ H_\delta^{n-1} (\partial A ) = \inf \left\{ \sum_{j=1}^{\infty} \alpha_{n-1}\frac{1}{2^{n-1}} [\operatorname{diam}(U_j)] ^{n-1} ...
2
votes
2answers
53 views
How smooth is the distribution function of a convex polynomial?
Here is a prototype of the problem I have in mind: Let $P:\mathbb{R}^2\rightarrow\mathbb{R}$ be a strictly convex, nonnegative polynomial such that $P(0,0)=0$. Let $\alpha\geq 0$, and consider the ...
20
votes
1answer
375 views
Which sets are removable for holomorphic functions?
[Note: I received a version of this question via email and decided to answer it on MSE, where it might be useful to others.]
Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some ...
2
votes
1answer
89 views
Equality in the Isoperimetric Inequality
Stein and Shakarchi, in their book Real Analysis, the third volume of the Princeton Lectures in Analysis series, give a proof of the isoperimetric inequality for closed rectifiable curves in ...
3
votes
1answer
81 views
Convolution square root of a Frostman measure
A probability measure $\mu$ on $\mathbb{R}^d$ is said to be a Frostman measure if
$$\mu(B)\lesssim r(B)^\alpha \ \ \ \ (1)$$
for all open ball $B$, where $r(B)$ denotes the radius and $\alpha>0$. ...
6
votes
0answers
92 views
Higher Order Coarea Formula
I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
1
vote
2answers
61 views
Is critical Haudorff measure a Frostman measure?
Let $K$ be a compact set in $\mathbb{R}^d$ of Hausdorff dimension $\alpha<d$, $H_\alpha(\cdot)$ the $\alpha$-dimensional Hausdorff measure. If $0<H_\alpha(K)<\infty$, is it necessarily true ...
2
votes
0answers
87 views
Integral Geometry Reference Request
I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
