Articles of geometric measure theory

Hausdorff measure of $f(A)$ where $f$ is a Holder continuous function.

Let $f\colon \mathbb R^d\to \mathbb R^k$ be a $\beta-$ Holder continuous function ($\beta \in (0,1)$) and $A\subset \mathbb R^d$. As for a Lipschitz function $g$ it holds that $H^s(g(A))\leq Lip(g)^s H^s(A)$, also for $f$ should hold a similar inequality. I want to know if what I’ve done is correct. So, let $f$ be such that […]

Prove Property of Doubling Measure on $\mathbb{R}$

Problem. Let $\mu$ be a fixed finite measure on $\mathbb{R}$. $\mu$ is said to be doubling if there exists a constant $C>0$ such that for any two adjacent intervals $I=[x-h,x]$ and $J=[x,x+h]$, $$\dfrac{\mu(I)}{C}\leq\mu(J)\leq C\mu(I)\tag{1}$$ Assuming that $\mu$ is doubling, show that there exist constants $C>0,\delta>0$ such that for every interval $I$, $$\mu(I)\leq B\left|I\right|^{\delta}\tag{2},$$ where $\left|\cdot\right|$ […]

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 $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ Conversely, suppose that the measurable sets $A_1,A_2,…$ satisfy $$\mu\left(\cup_{k=1}^\infty A_k\right) = \sum_{k=1}^\infty\mu(A_k)<\infty$$ Prove that the sets are almost disjoint. Here $\mu(A)$ denotes the Lebesgue measure of $A$. I know that if […]

Volume form and Hausdorff measure

Let $M$ be a smooth orientable $(n-1)$-dimensional submanifold in $\mathbb{R}^n$, $dS$ be its volume form and $dH^{n-1}(x)$ be an $(n-1)$-dimensional Hausdorff measure. How to show than that $$ \int\limits_{M} f(x) dS = \int\limits_{M} f(x) dH^{n-1}(x) $$ In fact, it is a generaliation of an equality formula for surface integrals of first and second kind in […]

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 $ f $ $ \int_M f d\mu = \sum_i^n \int_{U_{i}}(\rho_i\sqrt{G_i}f)\circ \phi_i^{-1}dx $ where $ (U_i,\phi_i ) $ is a finite […]

Change of variable within an integral of the Hausdorff measure

Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, $H^{m}$ be a Hausdorff measure. Is it true that $$ \int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx) $$ where $f(x)$ is some continuous function?

Filling a unit cube with countable balls.

Is it possible to fill an $n-$dimensional unit cube with countable number of non-overlapping $n-$dimensional balls? By $n-$dimensional unit cube, I am thinking of the set $$ C:=\{\ x\in\Bbb R^n\ |\ 0\le x_i\le 1\ \text{for all}\ i=1,2,\dots,n\ \} $$ where $x=(x_1,\dots,x_n)$. An $n-$dimensional ball centered at $x_0$ is defined to be $$ B(x_0;r):=\{\ x\in\Bbb R^n\ […]

metric and measure on the projective space

Let $RP^n$ be the $n$-dim real projective space. I have the following four questions. What is the so called standard metric on $RP^n$? More generally, consider a metric space $M$ with an equivalent relation ~. Then there is a natural way to define a (pseudo)metric on the quotient space $M$\~. See Why are quotient metric […]

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove or disprove that $$ \sup \left\{\frac{\int_E |f|dx}{|E|^{1/2}}: E \subset [0,1]\right\}<+\infty$$ If the claim above is false, then is it possible to prove that for any fixed […]

Can a $N-1$ rectifiable set be partitioned into countably many connected pieces?

Let $\Gamma\subset \mathbb R^N$ be a $N-1$ rectifiable curve such that $\mathcal H^{N-1}(\Gamma)<\infty$. I am wondering that would it be possible to partition it into countably many connection pieces, up to $\epsilon>0$ error? (where $\epsilon>0$ is given) That is, would it be possible to find a set $\{\Gamma_n\}_{n=1}^\infty$ such that each $\Gamma_n\subset \Gamma$ is connected, […]