1
vote
0answers
27 views
+50

Measurability of upper and lower derivatives of Radon measures

Let $\mu$ and $\nu$ be Radon measures in $\mathbb R^N$. Define their upper and lower derivatives by $$ \overline{D}_\nu\mu(x):=\limsup_{r\to0}\frac{\mu(B_r(x))}{\nu(B_r(x))},\qquad ...
0
votes
0answers
20 views

Borel outer measure and Borel measure

I have these two definitions: Given $\ (X,\mathcal{M},\mu)$ measure space, the measure $\ \mu$ is Borel measure if $\ \mathcal{M}=\mathcal{B}(X) $ Given $\ \phi:X\to[-\infty,+\infty]$ be an outer ...
0
votes
1answer
37 views

Indicator function of a level set

Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$. Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
3
votes
4answers
125 views

Prove that $\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$

Suppose that the measurable sets $A_1,A_2,...$ are "almost disjoint" in the sense that $\mu(A_i\cap A_j) = 0$ if $i\neq j$. Prove that $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ ...
6
votes
1answer
133 views

A curious problem about Lebesgue measure.

The Problem: Let $(B(x_{m},0.5))_{m}$ be a sequence of disjoint open discs in $\mathbb{R}^{2}$ centered in $x_{m}$ and with radius 0.5. Let $\psi(n)$ be the number of these discs contained in the ...
2
votes
0answers
57 views

Measures whose projections are absolutely continuous

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely ...
2
votes
1answer
90 views

Two Definitions of Minkowski Dimension

I'm currently reading a paper. Let $F\subset\mathbb R^n$ and $\epsilon\gt0$, the paper defined $m^s(F):=\liminf_{\epsilon \to 0}\epsilon^{s-n}\lambda(F_\epsilon)$ and $M^s(F):=\limsup_{\epsilon \to ...
1
vote
1answer
117 views

Simple questions about Hausdorff measure

Let $s>0$ and $0 < \delta \leq \infty$. For a set $E \subset R^n$ define $$ H_{s}^{\delta} (E)=\inf \left\{ \sum_i r_{i}^{s}\right\},$$ where the infumum is taken over all coverings of $E$ by ...
2
votes
1answer
114 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 ...
2
votes
1answer
133 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 ...
1
vote
0answers
123 views

Change of variables formula for a general measure

In a paper, pp11, I read the equality ...
1
vote
1answer
142 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 ...
1
vote
1answer
80 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 ...
8
votes
1answer
163 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.
6
votes
1answer
272 views

Is the Hausdorff outer measure regular?

An outer measure $\mu^*$ is said to be regular if for every set $A \subset X$ $$\mu^\ast (A)=\inf\{\mu^*(E) : E\supset A \text{ is } \mu^\ast\text{-measurable} \}$$ To check that an outer ...
0
votes
1answer
104 views

Reference for Radon measure

Can anyone provide references for Radon measures, Bounded variation functions space, and Lebesgue differentiation theorem for Radon Measures, Hausdorff dimensions? Note: Online references will be ...