Geometry of Measures in Real Dimensions via Hölder Parameterizations

Abstract

We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb {R}^n$ when s is a real number between 0 and n. This topic in geometric measure theory has been extensively studied when s is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on s-sets by Martín and Mattila from 1988 to 2000. When $0<s<1$, we prove that measures with almost everywhere positive lower density and finite upper density are carried by countably many bi-Lipschitz curves. When $1\le s<n$, we identify conditions on the lower density that ensure the measure is either carried by or singular to (1 / s)-Hölder curves. The latter results extend part of the recent work of Badger and Schul, which examined the case $s=1$ (Lipschitz curves) in depth. Of further interest, we introduce Hölder and bi-Lipschitz parameterization theorems for Euclidean sets with “small” Assouad dimension.

Appendix A: Decomposition of \(\sigma \)-Finite Measures

The following definition encodes commonly used definitions of countably rectifiable and purely unrectifiable measures, including the variants in Definition 1.1.

Definition A.1

Let $(\mathbb {X},\mathcal {M})$ be a measurable space, let $\mathcal {N}\subseteq \mathcal {M}$ be a non-empty collection of measurable sets, and let $\mu$ be a measure defined on $(\mathbb {X},\mathcal {M})$. We say that $\mu$ is carried by$\mathcal {N}$ provided there exists a countable family $\{\Gamma _i:i\ge 1\}\subseteq \mathcal {N}$ of sets with

$$\begin{aligned} \mu \left( \mathbb {X}\setminus \bigcup _{i=1}^\infty \Gamma _i\right) =0. \end{aligned}$$

We say that $\mu$ is singular to$\mathcal {N}$ if $\mu (\Gamma )=0$ for every $\Gamma \in \mathcal {N}$.

The “correctness” of Definition A.1 is partially justified by the following proposition, which should be considered a standard exercise in measure theory. The proof is a slight variation of [10, Proposition 1.1] (or [30, Theorem 15.6]), which is specialized to the decomposition of Radon measures (sets) in $\mathbb {R}^n$ into countably m-rectifiable and purely m-unrectifiable components. We present details for the convenience of the reader.

Proposition A.2

(Decomposition) Let $(\mathbb {X},\mathcal {M})$ be a measurable space and let $\mathcal {N}\subseteq \mathcal {M}$ be a non-empty collection of sets. If $\mu$ is a $\sigma$-finite measure on $(\mathbb {X},\mathcal {M})$, then $\mu$ can be written uniquely as

$$\begin{aligned} \mu =\mu _\mathcal {N}+\mu _\mathcal {N}^\perp ,\end{aligned}$$

(A.1)

where $\mu _\mathcal {N}$ is a measure on $(\mathbb {X},\mathcal {M})$ that is carried by $\mathcal {N}$ and $\mu ^\perp _\mathcal {N}$ is a measure on $(\mathbb {X},\mathcal {M})$ that is singular to $\mathcal {N}$.

Proof

Let $\widetilde{\mathcal {N}}$ denote the collection of finite unions of sets in $\mathcal {N}$. Given a $\sigma$-finite measure $\mu$ on $(\mathbb {X},\mathcal {M})$, expand $\mathbb {X}=\bigcup _{j=1}^\infty X_j$, where

$$\begin{aligned} X_1\subseteq X_2\subseteq \cdots \end{aligned}$$

is an increasing chain of sets in $\mathcal {M}$ with $\mu (X_j)<\infty$ for all $j\ge 1$. For each $j\ge 1$, define

$$\begin{aligned} M_j := \sup _{N\in \widetilde{\mathcal {N}}} \mu (X_j\cap N)\le \mu (X_j)<\infty . \end{aligned}$$

By the approximation property of the supremum, we may choose a sequence $(N_j)_{j=1}^\infty$ of sets in $\widetilde{\mathcal {N}}$ such that $\mu (X_j\cap N_j)>M_j-1/j$ for all $j\ge 1$. Fix any such $(N_j)_{j=1}^\infty$ and define

Then $\mu _\mathcal {N}$ and $\mu ^\perp _\mathcal {N}$ are measures on $(\mathbb {X},\mathcal {M})$ with $\mu =\mu _\mathcal {N}+\mu _\mathcal {N}^\perp$ and it is clear that $\mu _{\mathcal {N}}$ is carried by $\mathcal {N}$.

To see that $\mu ^\perp _\mathcal {N}$ is singular to $\mathcal {N}$, assume for contradiction that $\mu ^\perp _{\mathcal {N}}(S)>0$ for some $S\in \mathcal {N}$. First pick an index $j_0$ such that $\mu (X_{j_0}\cap S)>0$. Next, pick $j\ge j_0$ sufficiently large such that $\mu (X_{j_0}\cap S)>1/j$. Note that $T:=N_j\cup S\in \widetilde{\mathcal {N}}$, since $N_j\in \widetilde{\mathcal {N}}$ and $S\in \mathcal {N}$. It follows that

$$\begin{aligned} M_j\ge \mu (X_j\cap T) \ge \mu _\mathcal {N}(X_j\cap N_j) + \mu ^\perp _\mathcal {N}(X_j\cap S)>(M_j-1/j)+1/j=M_j, \end{aligned}$$

where in the last inequality we used the fact that $X_{j_0}\subseteq X_j$. We have a reached a contradiction. Therefore, $\mu ^\perp _\mathcal {N}$ is singular to $\mathcal {N}$.

Next we want to show that the decomposition of $\mu$ as the sum of a measure that is carried by $\mathcal {N}$ and a measure that is singular to $\mathcal {N}$ is unique. Suppose that $\mu =\mu _c+\mu _s$, where $\mu _c$ and $\mu _s$ are measures such that $\mu _c$ is carried by $\mathcal {N}$ and $\mu _s$ is singular to $\mathcal {N}$. To show that $\mu _c=\mu _\mathcal {N}$ and $\mu _s=\mu ^\perp _\mathcal {N}$, it suffices to prove the former. Suppose for contradiction that $\mu _c(A)<\mu _\mathcal {N}(A)$ for some $A\in \mathcal {M}$. Replacing A with $A\cap X_j$ for j sufficiently large, we may assume without loss of generality that $\mu _{\mathcal {N}}(A)<\infty$. Since $\mu _c$ and $\mu _\mathcal {N}$ are both carried by $\mathcal {N}$, we can find a set N, which is a countable union of sets in $\mathcal {N}$ such that

$$\begin{aligned} \mu _c(A\cap N)=\mu _c(A)<\mu _\mathcal {N}(A)=\mu _\mathcal {N}(A\cap N). \end{aligned}$$

Then $\mu _s(A\cap N)=\mu (A\cap N)-\mu _c(A\cap N) > \mu (A\cap N)-\mu _\mathcal {N}(A\cap N)=\mu ^\perp _\mathcal {N}(A\cap N)=0.$ This contradicts that $\mu _s$ is singular to $\mathcal {N}$. Therefore, $\mu _c=\mu _\mathcal {N}$, and thus, $\mu _s=\mu ^\perp _\mathcal {N}$.$\square$

Example A.3

Let $\mu$ and $\nu$ be measures on a measurable space $(\mathbb {X},\mathcal {M})$, and let

$$\begin{aligned} \mathcal {N}:=\{A\in \mathcal {M}:\nu (A)=0\} \end{aligned}$$

denote the null sets of $\nu$. If $\mu$ is $\sigma$-finite, then by Proposition A.2, the measure $\mu$ can be uniquely expanded $\mu =\mu _\mathcal {N}+\mu _{\mathcal {N}}^\perp$, where $\mu _{\mathcal {N}}$ is carried by null sets of $\nu$ and $\mu _{\mathcal {N}}^\perp$ is singular to null sets of $\nu$. Thus, writing $\mu _s:=\mu _{\mathcal {N}}$ and $\mu _{ac}:=\mu _{\mathcal {N}}^\perp$, we can decompose $\mu =\mu _{s}+\mu _{ac}$, where $\mu _{s}\perp \nu$ and $\mu _{ac}\ll \nu$.