Generalized rectifiability of measures and the identification problem
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Abstract
One goal of geometric measure theory is to understand how measures in the plane or a higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz images of subspaces or Lipschitz graphs), but all of them can be encoded using the same framework. In this paper, we describe a framework for generalized rectifiability, review a selection of classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz or Hölder or \(C^{1,\alpha }\) images of Euclidean subspaces, including theorems of Azzam–Tolsa, Badger–Schul, Badger–Vellis, Edelen–Naber–Valtorta, Ghinassi, and Tolsa–Toro.
Keywords
Structure of measures Atoms Generalized rectifiability Fractional rectifiability Density ratios Flatness Geometric square functionsMathematics Subject Classification
Primary 28A75 Secondary 26A16 42B99 54F501 Introduction
Given a measure, perhaps one of the most fundamental problems is to determine which sets have positive measure and which sets have zero measure. In this paper, we are interested in a dual problem: given a class of sets, we want to determine which measures assign all of their mass to those sets and which measures vanish on each of those sets. Special cases of the dual problem are commonly studied in geometric measure theory, under the heading of rectifiability of measures. To formally state this (see Problem 1.7), we need to first introduce some terminology (see Definition 1.1), which seems to be missing from the standard lexicon. Recall that a measurable space \(({\mathbb {X}},{\mathcal {M}})\) is a nonempty set \({\mathbb {X}}\), equipped with a \(\sigma\)algebra \({\mathcal {M}}\), i.e., a nonempty family of subsets of \(\mathbb {X}\) that is closed under taking complements and countable unions. By measure, we mean a positive measure, i.e., a function \(\mu :{\mathcal {M}}\rightarrow [0,\infty ]\) with \(\mu (\emptyset )=0\) that is countably additive on disjoint sets.
Definition 1.1
 (1)
\(\mu\) is carried by \({\mathcal {N}}\) if there exist countably many \(N_i\in {\mathcal {N}}\) such that \(\mu ({\mathbb {X}}\setminus \bigcup _i N_i)=0\);
 (2)
\(\mu\) is singular to \({\mathcal {N}}\) if \(\mu (N)=0\) for every \(N\in {\mathcal {N}}\).
Example 1.2
A validation of Definition 1.1 is that every \(\sigma\)finite measure can be uniquely written as the sum of a measure carried by \({\mathcal {N}}\) and a measure singular to \({\mathcal {N}}\). In the statement of Proposition 1.3 and below, we let \(\mu \,\!\_\,A\) denote the restriction of a measure \(\mu\) to a measurable set A; i.e., \(\mu \,\!\_\,A\) is the measure defined by the rule \((\mu \,\!\_\,A)(B)=\mu (A\cap B)\) for all measurable sets B. A measure is \(\sigma\)finite if \({\mathbb {X}}=\bigcup _{i=1}^\infty X_i\) for some sets \(X_i\in {\mathcal {M}}\) with \(\mu (X_i)<\infty\) for all i.
Proposition 1.3
Proof
Example 1.4
Example 1.5
Example 1.6
Although Proposition 1.3 provides for the decomposition of any \(\sigma\)finite measure into component measures carried by or singular to \({\mathcal {N}}\), the proof of this fact is abstract (as it relies on the completeness axiom of \(\mathbb {R}\) and the approximation property of the supremum) and does not provide a concrete method to identify the components for a particular measure. This leads us to the following problem, which is our main problem of interest.
Problem 1.7
There is room for debate on what constitutes a “good” solution of Problem 1.7, but in the author’s view a reasonable solution should generally involve the geometry of the space \({\mathbb {X}}\) or sets \({\mathcal {N}}\). If this includes the ability to sample a measure on a ball, then the atomic identification problem for locally finite measures in a metric space is easily solved.
Example 1.8
Here the restriction to locally finite measures is crucial. For example, if \(\{\ell _i\}_{i=1}^\infty\) is an enumeration of straight lines in the plane that pass through the origin and have rational slopes, then \(\mu =\sum _{i=1}^\infty {\mathscr {L}}^1\,\!\_\,\ell _i\) is \(\sigma\)finite and atomless, but \(\mu (B(0,r))=\infty\) for all \(r>0\).
The identification problem for 1rectifiable measures was first studied by Besicovitch [6, 7] in a broader investigation into the geometry of planar sets with positive and finite length, and later by Morse and Randolph [8], Moore [9], Pajot [10], Lerman [11], and Azzam and Mourgoglou [12]. Complete solutions within the classes of pointwise doubling measures and Radon measures in \(\mathbb {R}^n\) were furnished very recently by Badger and Schul [13]. A description of the latter will be presented in Sect. 2.
Example 1.9
Example 1.10
Example 1.11
Example 1.12
The examples above illustrate different solutions of Problem 1.7 when \({\mathcal {N}}\) is the collection of rectifiable curves in \(\mathbb {R}^n\) and \({\mathscr {F}}\) is one of several sets of \(\sigma\)finite Borel measures on \(\mathbb {R}^n\). Additional results are available when \({\mathcal {N}}\) is the collection of images of Lipschitz maps \(f:[0,1]^m\rightarrow \mathbb {R}^n\) and \(\mu \ll \mathcal {H}^m\) (see Sect. 3), or when \(\mathbb {X}={\mathbb {H}}^n\) is the nth Heisenberg group (see Sect. 4), but in general we currently know far less than one should like. For example, even the following deceptively simple problem is presently open (cf. Example 1.2).
Problem 1.13
Let \({\mathcal {N}}\) denote the set of lines (1dimensional affine subspaces) in \(\mathbb {R}^2\). Identify the Radon measures on \(\mathbb {R}^2\) that are carried by \({\mathcal {N}}\) or singular to \({\mathcal {N}}\).
The rest of this survey is organized, as follows. In Sect. 2, we present the solution of the identification problem for 1rectifiable Radon measures from [13]. In Sect. 3, we review the current state of affairs on identification problems for mrectifiable measures when \(m\ge 2\). In Sect. 4, we discuss further directions and open problems on generalized rectifiability, including fractional rectifiability and higherorder rectifiability in \(\mathbb {R}^n\)—and other spaces.
2 Solution of the identification problem for 1rectifiable measures
Theorem 2.1

the lower 1dimensional Hausdorff density \({{\underline{D}}^{1}}(\mu ,\cdot )\) of a measure \(\mu\) (a common notion in geometric measure theory); and

a densitynormalized Jones function \(J^*_2(\mu ,\cdot )\) associated to “anisotropic” \(L^2\) beta numbers \(\beta ^*_2(\mu ,\cdot )\) (which are the main innovation in [13]).
Lemma 2.2
Proof
This is an easy consequence of the relationship between pointwise control on the lower density \({{\underline{D}}^{1}}(\mu ,\cdot )\) along a set E and the 1dimensional packing measure \({\mathcal {P}}^1\) of E (see Taylor and Tricot [32]); the finiteness of the packing measure \({\mathcal {P}}^1\) on bounded sets in \(\mathbb {R}\); and the interaction of packing measures with Lipschitz maps. See [24] for details.\(\square\)
Remark 2.3
Before we define the anisotropic beta number \(\beta ^*_2(\mu ,Q)\) from [13], we give two lemmas in order to motivate its definition. The following observation, an elegant application of Jensen’s inequality, is due to Lerman [11]. Practically speaking, it allows one to control the distance from the \(\mu\) center of mass of a window Q to a straight line \(\ell\) in terms of the quantity \(\beta _2(\mu ,Q,\ell )\).
Lemma 2.4
Proof
Given two windows R and Q with \(R\subset Q\), the approximation number \(\beta _2(\mu ,Q)\) controls the approximation number \(\beta _2(\mu ,R)\) if one has control on \(\mathrm {diam}\,Q / \mathrm {diam}\,R\) and \(\mu (Q) / \mu (R)\).
Lemma 2.5
Proof
This is immediate from monotonicity of the integral. \(\square\)

\(\mathrm {diam}\,R=\mathrm {diam}\,Q\) or \(\mathrm {diam}\,R=\frac{1}{2}\mathrm {diam}\,Q\); and

the concentric dilate 3R is contained in the concentric dilate \(1600\sqrt{n} Q\).
Theorem 2.6
Remark 2.7
Theorem 2.8
Theorem 2.9
(see [13, §5]) Let \(\mu\) be a Radon measure on \(\mathbb {R}^n\). Then we have \(\mu \,\!\_\,\left\{ x: J_2^{**}(\mu ,x)<\infty \right\}\) is 1rectifiable. (Hence \(\mu \,\!\_\,\left\{ x: J_2^{**}(\mu ,x)<\infty \right\} \le \mu ^1_{rect}\).)
Remark 2.10
The resolution of the identification problem for 1rectifiable Radon measures provides a template for attacking similar problems. To solve the identification problem for Radon measures carried by a family of sets \({\mathcal {N}}\) in a space \(\mathbb {X}\), there are three basic steps. First, find a suitable characterization of subsets of the sets in \({\mathcal {N}}\). Second, transform the result for sets into a characterization of doubling measures carried by \({\mathcal {N}}\). Third, introduce anisotropic normalizations to promote the characterization for doubling measures to a characterization for Radon measures. For example, I expect it should be possible to follow this plan to solve the identification problem for Radon measures in \(\mathbb {R}^n\) that are carried by mdimensional Lipschitz graphs.
3 Recent progress on higher dimensional rectifiability
 (1)
\(\mu\) is (Lipschitz) image mrectifiable if \(\mu\) is carried by images of Lipschitz maps \(f:[0,1]^m\rightarrow \mathbb {R}^n\);
 (2)
\(\mu\) is (Lipschitz) graph mrectifiable if \(\mu\) is carried by isometric copies of graphs of Lipschitz maps \(g:\mathbb {R}^{m}\rightarrow \mathbb {R}^{nm}\);
 (3)
\(\mu\) is \(C^1\) mrectifiable if \(\mu\) is carried by mdimensional embedded \(C^1\) submanifolds of \(\mathbb {R}^n\).
Theorem 3.1
Suppose \(E\subset \mathbb {R}^n\) is \({\mathcal {H}}^m\) measurable and \(0<\mathcal {H}^m(E)<\infty\). If \(\mu =\mathcal {H}^m\,\!\_\,E\) is image or graph mrectifiable, then \(\mu\) is \(C^1\) mrectifiable. More generally, if \(\mu\) is a Radon measure on \(\mathbb {R}^n\) and \(\mu \ll \mathcal {H}^m\), then \(\mu\) is mrectifiable with respect to one of the definitions (1), (2), or (3) if and only if \(\mu\) is mrectifiable with respect to each of (1), (2), and (3).
Proof
For the first part, see Federer [4, 3.2.29]. The second part follows from the first with the aid of the Radon–Nikodym theorem.\(\square\)
Remark 3.2
It would be nice if there were a universal convention about which version of rectifiability (image, graph, or \(C^1\)) is the default definition. My own preference is that image rectifiability should be default and, henceforth, will say simply that a Radon measure \(\mu\) on \(\mathbb {R}^n\) is mrectifiable if \(\mu\) is image mrectifiable. There are three basic reasons. First, image rectifiability is the definition that is consistent with the convention that a 1rectifiable measure is a measure carried by rectifiable curves, which is used implicitly and explicitly in the work of Besicovitch [6, 7] and Morse and Randolph [8]. Second, in the monograph [4] that gave the field of geometric measure theory its name, Federer makes the definition that \(E\subset \mathbb {R}^n\) is countably \((\mu ,m)\) rectifiable if \(\mu \,\!\_\,E\) is image mrectifiable in the sense above. (That is, even though Federer defines rectifiability as a property of a set rather than as a property of a measure, he uses Lipschitz images in the definition.) Third, of the three definitions, image rectifiability is the most general. For a setting where graph rectifiability is the most natural definition, see [35].
For any Radon measure \(\mu\) on \(\mathbb {R}^n\) and integer \(1\le m\le n1\), let \(\mu =\mu ^m_{rect}+\mu ^m_{pu}\) denote the decomposition from Proposition 1.3, where \(\mu ^m_{rect}\) is carried by images of Lipschitz maps \(f:[0,1]^m\rightarrow \mathbb {R}\) and \(\mu ^m_{pu}\) is singular to images Lipschitz maps \(f:[0,1]^m\rightarrow \mathbb {R}^n\) in the sense of Definition 1.1. The following fundamental problem in geometric measure theory about the structure of measures in Euclidean space is wide open.
Problem 3.3
(identification problem for mrectifiable measures) Let \(2\le m\le n1\). Find geometric or measuretheoretic properties that identify \(\mu ^m_{rect}\) and \(\mu ^m_{pu}\) for every Radon measure \(\mu\) on \(\mathbb {R}^n\). (Do not assume that \(\mu \ll \mathcal {H}^m\).)
Lemma 3.4
Theorem 3.5
 (1)
\(\mu\) is mrectifiable (carried by Lipschitz images of \([0,1]^m\));
 (2)
there is a unique \(\mu\) approximate tangent mplane at \(\mu\)a.e. \(x\in \mathbb {R}^n\);
 (3)\(\mu\) is weakly mlinearly approximable at \(\mu\)a.e. \(x\in \mathbb {R}^n\), i.e., \({{\underline{D}}^{m}}(\mu ,x)>0\) and$$\begin{aligned} \mathrm {Tan}(\mu ,x)\subseteq \{c{\mathcal {H}}^m\,\!\_\,L:c>0, L\in G(n,m)\}\quad \text {at } \mu \text {a.e. } x\in \mathbb {R}^n; \end{aligned}$$
 (4)the mdimensional Hausdorff density of \(\mu\) exists and is positive \(\mu\)a.e.:$$\begin{aligned} \liminf _{r\downarrow 0} \frac{\mu (B(x,r))}{\omega _m r^m}=\limsup _{r\downarrow 0}\frac{\mu (B(x,r))}{\omega _m r^m}>0\quad \text {at } \mu \text {a.e. } x\in \mathbb {R}^n; \end{aligned}$$
 (5)\({{\underline{D}}^{m}}(\mu ,x)>0\) at \(\mu\)a.e. \(x\in \mathbb {R}^n\) and the coarse density ratios are asymptotically optimally doubling in the sense that$$\begin{aligned} \lim _{r\downarrow 0}\frac{\mu (B(x,2r))}{\omega _m(2r)^m}\frac{\mu (B(x,r))}{\omega _m r^m}=0\quad \text {at } \mu \text {a.e. } x\in \mathbb {R}^n; \end{aligned}$$
 (6)\({{\overline{D}}^{\,m}}(\mu ,x)>0\) at \(\mu\)a.e. \(x\in \mathbb {R}^n\) and the Jones function associated to the mdimensional homogeneous \(L^2\) beta numbers \(\beta _2^{m,h}(\mu ,x,r)\) is finite \(\mu\)a.e.:$$\begin{aligned} \int _0^1 \beta _2^{m,h}(\mu ,x,r)^2\frac{\mathrm{d}r}{r}<\infty \quad \text {at } \mu \text {a.e. }x\in \mathbb {R}^n. \end{aligned}$$
Proof
The equivalence of (5) and (6) with (1), (2), (3), and (4) are recent developments. The implication \((4)\Rightarrow (5)\) is trivial. The implication \((5)\Rightarrow (4)\) was proved by Tolsa and Toro [40]. The implication \((1)\Rightarrow (6)\) was proved by Tolsa [41]. With the additional assumption \({{\underline{D}}^{m}}(\mu ,x)>0\) \(\mu\)a.e., the implication \((6)\Rightarrow (1)\) was proved by Pajot [10] (for Radon measures of the form \(\mu =\mathcal {H}^m\,\!\_\,E\), E compact) and Badger and Schul [25]. With the a priori weaker assumption \({{\overline{D}}^{\,m}}(\mu ,x)>0\) \(\mu\)a.e., the implication \((6)\Rightarrow (1)\) was proved by Azzam and Tolsa [42]. To obtain the last result, Azzam and Tolsa carry out an intricate stopping time argument using David and Mattila’s “dyadic” cubes [43] and David and Toro’s extension of Reifenberg’s algorithm to sets with holes [44]. \(\square\)
For related work on rectifiability of absolutely continuous measures and the theory of mass transport, see Tolsa [45], Azzam, David, and Toro [46, 47], and Azzam, Tolsa, and Toro [48]. For the connection between rectifiability of sets and Mengertype curvatures, see Léger [49], Lerman and Whitehouse [50, 51], Meurer [52], and Goering [53]. For related results about discrete approximation and rectifiability of varifolds, see Buet [54].
In recent work [55], Edelen, Naber, and Valtorta provide new sufficient conditions for qualitative and quantitative rectifiability of Radon measures \(\mu\) on \(\mathbb {R}^n\) that do not require absolute continuity, \(\mu \ll \mathcal {H}^m\) (\(\Leftrightarrow {{\overline{D}}^{\,m}}(\mu ,x)<\infty\) at \(\mu\)a.e. \(x\in \mathbb {R}^n\).) For simplicity, we state a special case of their main result.
Theorem 3.6
Remark 3.7
Edelen, Naber, and Valtorta’s proof of Theorem 3.6 is based on an updated, quantitative version of the Reifenberg algorithm (cf. [44] and [33]), the original version of which allows one to parameterize sets which are sufficiently “locally flat” at all locations and scales. A second proof of Theorem 3.6 has been provided by Tolsa [56] using a method closer to that of [42]. It is evident that the sufficient condition for a measure to be mrectifiable in Theorem 3.6 is not necessary in view of the example of Martikainen and Orponen [26] from the case \(m=1\).
Proposition 3.8
(see [59, §9.2]; “Cantor ladders”) For all \(m\ge 2\), there exist compact, connected, locally connected, mAhlfors regular sets \(G\subset \mathbb {R}^n\) such that G is not contained in the image of any Lipschitz map \(f:[0,1]^m\rightarrow \mathbb {R}^{m+1}\).
(Proof Sketch (\(m=2\))) Start with the set K above. One can transform K into a locally connected set G by adjoining sufficiently many squares \(S_{j,k}\times \{t_k\}\subset \mathbb {R}^2\times \mathbb {R}\), \(1\le j\le J_k\) on a countable dense set of heights \(t_k\in [0,1]\), with diameters of \(S_{j,k}\) vanishing as \(k\rightarrow \infty\). By carefully selecting parameters, one can ensure that G is Ahlfors 2regular. However, since G contains \(C\times [0,1]\) and \(\mathcal {H}^2\,\!\_\,(C\times [0,1])\) is purely 2unrectifiable, it is not possible that G is contained in the image of a Lipschitz map \(f:[0,1]^2\rightarrow \mathbb {R}^3\).\(\square\)
In order to attack Problem 3.3, it would be useful to first have new sufficient criteria for identifying Lipschitz images. In the author’s view, any solution of the following problem would be interesting (even if very far from a necessary condition).
Problem 3.9
For each \(m\ge 2\), find sufficient geometric, metric, and/or topological conditions that ensure a set \(\Gamma \subset \mathbb {R}^n\) is (contained in) a Lipschitz image of \([0,1]^m\).
Lipschitz images of \([0,1]^m\) represent only one possible choice of model sets to build a theory of higher dimensional rectifiability, and it may be worthwhile to explore other families of sets for which it is possible to solve Problem 1.7. One promising alternative is a class of surfaces that support a traveling salesman type theorem, recently identified by Azzam and Schul [60], which are lower regular with respect to the mdimensional Hausdorff content \(\mathcal {H}^m_\infty (E):=\inf \{\sum (\mathrm {diam}\,E_i)^m:E\subset \bigcup _i E_i\}\). For a complete description, we refer the reader to [60]. Also see Villa [61], which characterizes the existence of approximate tangent mplanes of content lower regular sets.
4 Fractional rectifiability and other frontiers
4.1 Fractional rectifiability
Problem 4.1
(identification problem for fractional rectifiability) Let \(1\le m\le n1\) be integers and let \(s\in [m,n)\). Find geometric or measuretheoretic properties that identify \(\mu _{m\rightarrow s}\) and \(\mu _{m\rightarrow s}^\perp\) for every Radon measure \(\mu\) on \(\mathbb {R}^n\). ( Do not assume that \(\mu \ll \mathcal {H}^s\).)
Hausdorff measures on selfsimilar sets provide essential examples of rectifiable and purely unrectifiable behavior in fractional dimensions. For further results in this direction, see Martín and Mattila [66] and Rao and Zhang [67].
Theorem 4.2
Theorem 4.3
(see Remes [68]) Let \(S=\bigcup _1^kf_i(S)\subset \mathbb {R}^n\) be a selfsimilar set of Hausdorff dimension \(s\in [1,n)\) that satisfies the open set condition. If S is compact and connected, then S is a (1 / s)Hölder curve.
For Radon measures, extreme behavior of the lower and upper Hausdorff densities force a measure to carried by or singular to (1 / s)Hölder curves and (m / s)Hölder mcubes.
Theorem 4.4
 (1)
\(\mu \,\!\_\,\left\{ x:{{\underline{D}}^{s}}(\mu ,x)=0\right\}\) is singular to (m / s)Hölder mcubes;
 (2)\(\nu\) is carried by (1 / s)Hölder curves, where$$\begin{aligned} \nu \equiv \mu \,\!\_\,\left\{ x:\int _0^1 \frac{r^s}{\mu (B(x,r))}\frac{\mathrm{d}r}{r}<\infty \text { and }\limsup _{r\downarrow 0}\frac{\mu (B(x,2r))}{\mu (B(x,r))}<\infty \right\} ; \end{aligned}$$
 (3)
\(\rho \equiv \mu \,\!\_\,\{x:0<{{\underline{D}}^{t}}(\mu ,x)\le {{\overline{D}}^{\,t}}(\mu ,x)<\infty \}\) is carried by (m / s)Hölder mcubes;
 (4)
moreover, if \(0\le t<1\), then \(\rho\) is carried by biLipschitz embeddings \(f:[0,1]\rightarrow \mathbb {R}^n\).
Example 4.5
The following problem is open, but from the point of view of topological dimension may be more tractable than the corresponding problem for Lipschitz squares (see Problem 3.9).
Problem 4.6
For all \(1<s<2\), find sufficient geometric, metric, and/or topological conditions that ensure a set \(\Gamma \subset \mathbb {R}^n\) is (contained in) a (1 / s)Hölder curve.
4.2 Higherorder rectifiability
Let \(1\le m\le n1\) and \(k\ge 1\) be integers, and let \(\alpha \in [0,1]\). We say that a Radon measure \(\mu\) on \(\mathbb {R}^n\) is \(C^{k,\alpha }\,m\) rectifiable if \(\mu\) is carried by mdimensional \(C^{k,\alpha }\) embedded submanifolds of \(\mathbb {R}^n\). In the case \(\alpha =0\), we also say that \(\mu\) is \(C^k\) mrectifiable. The study of higherorder rectifiability of measures was initiated by Anzellotti and Serapioni [69]. In general, different orders of rectifiability give rise to different classes of measures.
Theorem 4.7
 (1)
If \(k+\alpha < l+\beta\), then there exists an mset \(E\subset \mathbb {R}^n\) such that \(\mathcal {H}^m\,\!\_\,E\) is \(C^{k,\alpha }\) mrectifiable and purely \(C^{l,\beta }\) munrectifiable (i.e., singular to \(C^{l,\beta }\) submanifolds).
 (2)
If \(\mu\) is a Radon measure on \(\mathbb {R}^n\) and \(\mu \ll \mathcal {H}^m\), then \(\mu\) is \(C^{k,1}\) rectifiable if and only if \(\mu\) is \(C^{k+1}\) rectifiable.
Proof
For (1), see [69, Proposition 3.3]. When \(\mu =\mathcal {H}^m\,\!\_\,E\) and \(0<\mathcal {H}^m(E)<\infty\), (2) is a consequence of [4, 3.1.5]. The assertion for absolutely continuous measures then follows via the Radon–Nikodym theorem. Compare to Theorem 3.1. \(\square\)
 \(Q_xM=\{(y,A(y,y)):y\in T_x M\}\) for some bilinear symmetric form$$\begin{aligned} A:T_xM \times T_x M \rightarrow (T_xM)^\perp , \end{aligned}$$
 \(\mathcal {H}^m\,\!\_\,\xi _{x,\rho }(M)\) converges to \(\mathcal {H}^m\,\!\_\,Q_x M\) weakly in the sense of Radon measures as \(\rho \downarrow 0\), where \(\xi _{x,\rho }\) denotes the nonhomogeneous dilation of M at x defined by$$\begin{aligned} \xi _{x,\rho }:\mathbb {R}^n\rightarrow \mathbb {R}^n,\quad \xi _{x,\rho }(y)= \rho ^{1}\pi _{T_xM}(yx) + \rho ^{2} \pi _{T_xM^\perp }(yx). \end{aligned}$$
Example 4.8
Anzellotti and Serapioni [69] give geometric characterizations of \(C^{1,\alpha }\) rectifiability when \(\alpha \in [0,1]\). In the case \(\alpha =1\), their result is the following.
Theorem 4.9
 (1)
the approximate tangent plane \(T_x M_j\) exists,
 (2)
the approximate tangent paraboloid \(Q_x M_j\) exists, and
 (3)(see [4, 2.9.12] for definition of approximate limits)$$\begin{aligned} \mathrm {ap}\,\limsup _{{\mathop {y\in M_j}\limits ^{y\rightarrow x}}} \dfrac{d(T_yM_j,T_xM_j)}{yx}<\infty ,\quad d(T_1,T_2)\equiv \sup _{y=1} \pi _{T_1}(y)\pi _{T_2}(y). \end{aligned}$$
More recently, two new characterizations of \(C^{1,\alpha }\) rectifiability of absolutely continuous measures have been provided by Kolasiński [70] (using Mengertype curvatures) and by Ghinassi [71] (using \(L^2\) beta numbers), the latter of which can be stated as follows. See (3.1) for the definition of \(\beta _2^{m,h}(\mu ,x,r)\).
Theorem 4.10
It would be interesting to know whether Ghinassi’s sufficient condition is also necessary for \(C^{1,\alpha }\) rectifiability (compare to Theorem 3.5(6)). The following problem is also open.
Problem 4.11
(identification problem for \(C^{k,\alpha }\) mrectifiable measures) Find geometric or measuretheoretic properties that characterize \(C^{k,\alpha }\) mrectifiable measures when \(k\ge 2\). (To start, you may assume \(\mu \ll \mathcal {H}^m\).)
For recent progress on Problem 4.11 for Hausdorff measures, see Santilli [72].
4.3 Rectifiability in other spaces
Following Kirchheim [73], a metric space \((X,\rho )\) is called mrectifiable if the mdimensional Hausdorff measure on X is carried by images of Lipschitz maps from subsets of \(\mathbb {R}^m\) into X. When \(E\subset \mathbb {R}^n\), the space \((E,d_2)\) equipped with induced Euclidean metric is mrectifiable if and only if \(\mathcal {H}^m\,\!\_\,E\) is Lipschitz image rectifiable in the sense of §3. Kirchheim examined the structure of general mrectifiable metric spaces and proved that for those spaces with \(\mathcal {H}^m(X)<\infty\), the mdimensional Hausdorff density exists and equals 1 at \(\mathcal {H}^m\) almost every point.
Theorem 4.12
When \((X,\rho )=(E,d_2)\), where \(E\subset \mathbb {R}^n\) is equipped with the Euclidean metric and \(\mathcal {H}^m(E)<\infty\), the converse of Theorem 4.12 is also true. This was proved by Besicovitch [6] when \(m=1\) and \(n=2\) and by Mattila [36] for general m and n. The converse of Theorem 4.12 is also true when \(m=1\) for general metric spaces with \(\mathcal {H}^1(X)<\infty\) by Preiss and Tišer [14] (recall Example 1.9 above). In all other cases it is not presently known whether the converse of Kirchheim’s theorem is true or false.
Problem 4.13
Let \(m\ge 2\). Prove that for every metric space \((X,\rho )\) with \(\mathcal {H}^m(X)<\infty\) that (4.2) implies X is mrectifiable. Or find a counterexample.
Problem 4.13 is interesting even in the case when \((X,\rho )=(E,d_p)\) for some \(E\subset \mathbb {R}^n\) and \(d_p\) is the distance induced by the pnorm, \(p\ne 2\). For related work on existence of densities of measures in Euclidean spaces with respect to nonspherical norms, see the series of papers by Lorent [74, 75, 76].
Although a density only characterization of rectifiable metric spaces remains illusive, a metric analysis characterization of rectifiable spaces has recently been established by Bate and Li [77]. Lipschitz differentiability spaces were introduced by Cheeger [78] and examined in depth by Bate [79]. Roughly speaking, these are spaces that have a sufficiently rich curve structure to support a version of Rademacher’s theorem; we refer the reader to [79] for a detailed description and several characterizations of differentiability spaces. The following theorem is a simplified statement of Bate and Li’s main result.
Theorem 4.14

\((U,\rho ,\mathcal {H}^m)\) is an mdimensional Lipschitz differentiability space, and

\(0<{{\underline{D}}^{m}}(\mathcal {H}^m\,\!\_\,U,x)\le {{\overline{D}}^{\,m}}(\mathcal {H}^m\,\!\_\,U,x)<\infty\) at \(\mathcal {H}^m\)a.e. \(x\in U\).
A metric space \((X,\rho )\) is called purely m rectifiable if \(\mathcal {H}^m\,\!\_\,X\) is singular to images of Lipschitz maps from subsets of \(\mathbb {R}^m\) into X. For example, the (first) Heisenberg group \({\mathbb {H}}\) with topological dimension 3 and Hausdorff dimension 4 is purely munrectifiable for all \(m=2,3,4\) (see Ambrosio and Kirchheim [80, §7]). A notion of intrinsic rectifiability of sets in Heisenberg groups (i.e., rectifiability with respect to \(C^1\) images of homogeneous subgroups) was investigated by Mattila et al. [81]. For related developments, see [82] and [83]. A characterization of complete, purely munrectifiable metric spaces with \(\mathcal {H}^m(X)<\infty\) was recently announced by Bate [84]. A related sufficient condition for rectifiability inside a metric space (using Bate’s theorem) has been announced by David and Le Donne [85].
Notes
Author's contributions
The author was partially supported by NSF DMS Grants 1500382 and 1650546.
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