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.

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Notes

  1. 1.

    A related investigation on the Hausdorff dimension of projections of s-sets onto lower-dimensional subspaces was carried out earlier by Marstrand [27].

  2. 2.

    In fact, K is Ahlfors regular in the sense that \(\mathcal {H}^m(K\cap B(x,r))\sim r^m\) when \(x\in K\) and \(0<r\le \mathrm{diam}\,K\), because K is self-similar.

  3. 3.

    For the definition of and background on quasisymmetric maps, we refer the reader to [21].

  4. 4.

    Constructions of tree-like surfaces are by now classical. For instance, see [38, Figure 2.4.16].

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Correspondence to Matthew Badger.

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Badger was partially supported by NSF Grants 1500382 and 1650546. Part of this work was carried out while Badger attended the long program on Harmonic Analysis at MSRI in Spring 2017.

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

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 \(\mu _{ac}\ll \nu \).

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Badger, M., Vellis, V. Geometry of Measures in Real Dimensions via Hölder Parameterizations. J Geom Anal 29, 1153–1192 (2019). https://doi.org/10.1007/s12220-018-0034-2

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Keywords

  • Hölder parameterization
  • Assouad dimension
  • Uniformly disconnected sets
  • Geometry of measures
  • Hausdorff densities
  • Generalized rectifiability

Mathematics Subject Classification

  • Primary 28A75
  • Secondary 26A16, 30L05