Generalized rectifiability of measures and the identification problem
Matthew Badger


Generalized rectifiability of measures]Generalized rectifiability of measures and
 the identification problem
The author was partially supported by NSF DMS grants 1500382 and 1650546.

[2010]Primary 28A75. Secondary 26A16, 42B99, 54F50


Department of Mathematics
 University of Connecticut
 Storrs, CT 06269-3009
matthew.badger@uconn.edu


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,α images of Euclidean subspaces, including theorems of Azzam-Tolsa, Badger-Schul, Badger-Vellis, Edelen-Naber-Valtorta, Ghinassi, and Tolsa-Toro.

This survey paper is based on a talk at the Northeast Analysis Network Conference held in Syracuse, New York in September 2017.


[
    Matthew Badger
    January 20, 2019
====================









§ 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 <ref>), we need to first introduce some terminology (see Definition <ref>), which seems to be missing from the standard lexicon.
Recall that a measurable space (𝕏,ℳ) is a nonempty set 𝕏, equipped with a σ-algebra ℳ, i.e. a nonempty family of subsets of  that is closed under taking complements and countable unions. By measure, we mean a positive measure, i.e. a function μ:ℳ→[0,∞] with μ(∅)=0 that is countably additive on disjoint sets.

 Let (𝕏,ℳ) be a measurable space, let μ be a measure defined on ℳ, and let 𝒩⊆ℳ be a family of measurable sets. We say that 

  * μ is carried by 𝒩 if there exist countably many N_i∈𝒩 such that μ(𝕏∖⋃_i N_i)=0;

  * μ is singular to 𝒩 if μ(N)=0 for every N∈𝒩.



In a measurable metric space (𝕏,ℳ) such that  ℳ contains the Borel σ-algebra ℬ_, the support of a measure μ defined on ℳ is the closed set defined by 
    μ:={x∈:μ(B(x,r))>0 for all r>0}.
 Equivalently, the support of μ is the smallest closed set F⊆ such that μ(∖ F)=0.



 Let {x_i}_1^∞, {I_i}_1^∞, {S_i}_1^∞ be dense sequences of points, unit line segements, and unit squares in ^2 (see Figure <ref>), meaning that ⋃_i=1^∞{x_i}=⋃_i=1^∞ I_i=⋃_i=1^∞ S_i=^2, and let a_i>0 be a sequence of weights such that ∑_1^∞ a_i=1. Define three Borel measures μ_0, μ_1, and μ_2  on ^2 by 
    μ_0:=∑_1^∞ a_i δ_x_i,    μ_1:=∑_1^∞ a_i ℒ^1 I_i,    μ_2:=∑_1^∞ a_i ℒ^2 S_i,
 where δ_x_i is the Dirac measure at x_i, ℒ^1 I_i is 1-dimensional Lebesgue measure restricted to I_i, and ℒ^2 S_i is 2-dimensional Lebesgue measure restricted to  S_i. The measures μ_0, μ_1, and μ_2 are probability measures on ^2 with common support, 
    μ_0=μ_1=μ_2=^2.
 However, μ_0 is carried by 0-dimensional sets (points), μ_1 is carried by 1-dimensional sets (line segements), and μ_2 is carried by 2-dimensional sets (squares). On the other hand, μ_1 and μ_2 are singular to points and μ_2 is singular to lines. Thus, the three measures are distinguished by their underlying carrying sets. This shows that the support of a measure is a rough approximation that may hide underlying structure of a measure. Definition <ref> provides us with language to discuss this structure.




A validation of Definition <ref> is that every σ-finite measure can be uniquely written as the sum of a measure carried by 𝒩 and a measure singular to 𝒩. In the statement of Proposition <ref> and below, we let μ A denote the restriction of a measure μ to a measurable set A; i.e., μ A is the measure defined by the rule (μ A)(B)=μ(A∩ B) for all measurable sets B. A measure is σ-finite if 𝕏=⋃_i=1^∞ X_i for some sets X_i∈ℳ with μ(X_i)<∞ for all i.

  Let (𝕏,ℳ) be a measurable space and let 𝒩⊆ℳ be a family of measurable sets. If μ is a σ-finite measure defined on ℳ, then μ can be uniquely written as 
    μ=μ_𝒩+μ_𝒩^⊥,
 where μ_𝒩 and μ_𝒩^⊥ are measures defined on ℳ such that μ_𝒩 is carried by 𝒩 and μ_𝒩^⊥ is singular to 𝒩. Moreover, there exists A_μ∈ℳ (a countable union of sets in 𝒩) such that 
    μ_𝒩=μ A_μ,    μ_𝒩^⊥=μ (𝕏∖ A_μ).
 The measure μ is carried by 𝒩 if and only if μ=μ_𝒩; and, similarly, μ is singular to 𝒩 if and only if μ=μ_𝒩^⊥.

 This is an elementary exercise in measure theory. We present a portion of the proof in order to motivate the identification problem (see Problem <ref>). Assume that 𝕏=⋃_i=1^∞ X_i for some sequence X_i∈ℳ of measurable sets with μ(X_i)<∞. Let 𝒩 denote the collection of finite unions of sets in 𝒩, and for each j≥ 1, define 
    α_j:=sup_A∈𝒩μ((X_1∪…∪ X_j)∩ A) ≤∑_i=1^jμ(X_i)<∞.
 By the approximation property of supremum, for each j≥ 1, we can choose a set A_j∈𝒩 such that μ((X_1∪…∪ X_j)∩ A_j) ≥α_j - 1/j. We now define 
    μ_𝒩:=μ(⋃_j=1^∞ A_j),    μ_𝒩^⊥:=μ(𝕏∖⋃_j=1^∞ A_j).
 Clearly μ=μ_𝒩+μ_𝒩^⊥ and the measure μ_𝒩 is carried by 𝒩. We leave it to the reader to verify the measure μ_𝒩^⊥ is singular to 𝒩 and that the decomposition of μ as the sum of a measure carried by 𝒩 and a measure singular to 𝒩 is unique. A complete proof can be found in the appendix of <cit.>.

Let (𝕏,ℳ) be a measurable space. Let ν be a measure defined on ℳ and consider 𝒩:={A∈ℳ: ν(A)=0}. Applying Proposition <ref>, we recover the familiar fact that if μ is a σ-finite measure defined on ℳ, then 
    μ=σ+ρ,
 where σ:=μ_𝒩 and ρ:=μ_𝒩^⊥ satisfy σ⊥ν and ρ≪ν. In other words, we obtain the Lebesgue-Radon-Nikodym decomposition of a σ-finite measure μ as the sum of a singular measure and an absolutely continuous measure with respect to an arbitrary auxiliary measure ν.

Let (𝕏,ℳ) be a measurable space. Let 𝒩 denote the collection of atoms of ℳ; that is, A∈𝒩 if and only if A∈ℳ is nonempty and B⊊ A for some B∈ℳ implies B=∅. (In the common situtation that 𝕏 is a metric space and ℳ contains the Borel σ-algebra on 𝕏, the family of atoms of ℳ is precisely the set of singletons of 𝕏.) For every atom A, the Dirac measure δ_A is defined by the rule δ_A(B)=1, if A⊆ B, and δ_A(B)=0, otherwise. By Proposition <ref>, we obtain that for any σ-finite measure μ defined on ℳ, 
    μ=μ_∙+μ_∘,
 where μ_∙:=μ_𝒩 is atomic (carried by the atoms of ℳ) and μ_∘:=μ_𝒩^⊥ is atomless (singular to the atoms of ℳ). Moreover, in this situation, one can easily verify that either μ_∙=0 or there exist finitely or countably many atoms A_i and weights 0<c_i<∞ such that 
    μ_∙=∑_ic_i δ_A_i.


 Assume that (𝕏,ℳ)=(^n,ℬ_^n), where ℬ_^n denotes the Borel σ-algebra on ^n. A rectifiable curve in ^n is the image f([0,1]) of a continuous map f:[0,1]→^n of finite total variation, or equivalently, is the image of a Lipschitz continuous map f:[0,1]→^n (see e.g. <cit.>). Let 𝒩 denote the collection of all rectifiable curves in ^n. By Proposition <ref>, every σ-finite Borel measure μ on ^n can be uniquely written as 
    μ=μ^1_+μ^1_,
 where μ^1_:=μ_𝒩 is 1-rectifiable (carried by rectifiable curves) and μ^1_:=μ_𝒩^⊥ is purely 1-unrectifiable (singular to rectifiable curves). For instance, of the three measures in Example <ref>, the measures μ_0 and μ_1 are 1-rectifiable, while the measure μ_2 is purely 1-unrectifiable. Finding geometric and/or measure-theoretic properties that distinguish between 1-rectifiable and purely 1-unrectifiable measures and their higher dimensional variants constitutes a large program in geometric measure theory; see e.g. <cit.>, <cit.>, <cit.>. We will define and discuss m-rectifiable measures and purely m-unrectifiable measures with m>1 in 3. 






Although Proposition <ref> provides for the decomposition of any σ-finite measure into component measures carried by or singular to 𝒩, the proof of this fact is abstract (as it relies on the completeness axiom of  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.

[identification problem]  Let (𝕏,ℳ) be a measurable space, let 𝒩⊆ℳ be a family of measurable sets, and let ℱ be a family of σ-finite measures defined on ℳ. Find properties P(μ,x) and Q(μ,x) defined for all μ∈ℱ and x∈𝕏 such that 
    μ_𝒩 = μ{x∈𝕏: P(μ,x)  holds}  and  μ_𝒩^⊥ = μ{x∈𝕏: Q(μ,x)  holds}.
 In other words, find (pointwise) properties that identify the part of μ carried by 𝒩 and the part of μ singular to 𝒩 for all measures in the class ℱ.


There is room for debate on what constitutes a “good" solution of Problem <ref>, but in the author's view a reasonable solution should generally involve the geometry of the space 𝕏 or sets 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.

Let (𝕏,ℳ)=(𝕏,ℬ_𝕏), where 𝕏 is a metric space and ℬ_𝕏 denotes the Borel σ-algebra on 𝕏. Let 𝒩 be the collection of singletons in 𝕏 and let ℱ denote the collection of locally finite, σ-finite Borel measures. If μ∈ℱ, then 
    μ_∙≡μ_𝒩=μ{x∈𝕏: lim_r↓ 0μ(B(x,r))>0},


    μ_∘≡μ_𝒩^⊥=μ{x∈𝕏: lim_r↓ 0μ(B(x,r))=0}
 by continuity of measures from above.

Here the restriction to locally finite measures is crucial. For example, if {ℓ_i}_i=1^∞ is an enumeration of straight lines in the plane that pass through the origin and have rational slopes, then μ=∑_i=1^∞ℒ^1ℓ_i is σ-finite and atomless, but μ(B(0,r))=∞ for all r>0.


The identification problem for 1-rectifiable measures was first studied by Besicovitch <cit.> in a broader investigation into the geometry of planar sets with positive and finite length, and later by Morse and Randolph <cit.>, Moore <cit.>, Pajot <cit.>, Lerman <cit.>, and Azzam and Mourgoglou <cit.>. Complete solutions within the classes of pointwise doubling measures and Radon measures in ^n were furnished very recently by Badger and Schul <cit.>. A description of the latter will be presented in 2.

 Let (𝕏,ℳ)=(^2,ℬ_^2), let 𝒩 be the collection of rectifiable curves in ^2, and let ℱ:={^1 E:E∈ℬ_^2 and  0<^1(E)<∞}, where ^s denotes the s-dimensional Hausdorff measure defined by 
    ^s(E):=lim_δ↓ 0inf{∑_i=1^∞ ( E_i)^s: E⊆⋃_i=1^∞ E_i and  E_i≤δ}.
 Besicovitch <cit.> proved that if μ=^1 E∈ℱ, then 
    μ^1_≡μ_𝒩=^1{x∈ E: lim_r↓ 0^1(E∩ B(x,r))/2r=1},


    μ^1_≡μ_𝒩^⊥=^1{x∈ E: lim inf_r↓ 0^1(E∩ B(x,r))/2r≤3/4}.
 Besicovitch conjectured that the constant 3/4 may be replaced by 1/2, but finding the optimal constant is still an open problem. The best result to date is due to Preiss and Tišer <cit.>, who showed that 3/4 may be replaced by the constant (2+√(46))/12=0.731…; moreover, Preiss and Tišer showed this holds in any metric space. A special case of the 1/2 conjecture for sets with an a priori flatness condition was settled by Farag <cit.>.

In <cit.>, Besicovitch also showed that if μ=^1 E∈ℱ, then 
    μ^1_=^1{x∈ E:E has an ^1 approximate tangent line at x},


    μ^1_=^1{x∈ E: E does not have an ^1 approximate tangent line at x}.
 Here we say that E has an ^1 approximate tangent line at x if 
    lim sup_r↓ 0^1(E∩ B(x,r))/2r>0
 and there exists a line L containing x such that 
    lim sup_r↓ 0^1(E∩ B(x,r)∖ X(x,L,α))/2r=0  for all α∈(0,π/2],
 where X(x,L,α) denotes the cone of points y∈^2 such that the line {x+t(y-x):t∈} meets the line L at an angle at most α (see Figure 1.2). For a contemporary, self-contained presentation of these results, see <cit.>. 



Let (𝕏,ℳ)=(^2,ℬ_^2), let 𝒩 be the collection of rectifiable curves in ^2, and let ℱ denote the collection of Radon measures (locally finite, Borel regular measures) μ on ^2 such that μ≪^1. Morse and Randolph <cit.> proved that if μ∈ℱ, then 
    μ^1_≡μ_𝒩=μ{x∈^2: lim_r↓ 0μ(B(x,r))/2r exists, and 0<lim_r↓ 0μ(B(x,r))/2r<∞},


    μ^1_≡μ_𝒩^⊥=μ{x∈^2: lim inf_r↓ 0μ(B(x,r))/2r≤100/101lim sup_r↓ 0μ(B(x,r))/2r}.
 Moore <cit.> extended Morse and Randolph's density characterizations of μ^1_ and μ^1_ to Radon measures μ in ^n such that μ≪^1 for all dimensions n≥ 2. Higher dimensional analogues of the density characterization of absolutely continuous m-rectifiable measures in ^n were established by Preiss <cit.>.

In <cit.>, Morse and Randolph also proved that if μ is a Radon measure in ^2 such that μ≪^1, then 
    μ^1_=μ{x∈^n:there is a μ approximate tangent line at x},


    μ^1_=μ{x∈^n:  there is not a μ approximate tangent line at x},
 where μ approximate tangent lines are defined by analogy with ^1 E approximate tangent lines in an obvious way (replacing ^1 E with μ). Higher dimensional analogues of the tangential characterization of absolutely continuous m-rectifiable measures in ^n were supplied by Federer <cit.>.


 For any Radon measure μ on ^n, location x∈^n, and scale r>0, define the homogeneous 1-dimensional L^2 Jones beta number β^h_2(μ,x,r) by 
    β^h_2(μ,x,r)^2=inf_ℓ∫_B(x,r)((y,ℓ)/r)^2 dμ(y)/r,
 where the infimum runs over all straight lines (1-dimensional affine subspaces) in ^n. Roughly speaking, β^h_2(μ,x,r) records in an L^2 gauge how well the measure μ B(x,r) can be approximated by a measure supported on a line. Here homogeneous refers to the normalization of the measure in (<ref>)—see 2 below for further discussion. The number β^h_2(μ,x,r) is one instance of a larger family of unilateral approximation numbers that originated in work of Jones <cit.>, Okikolu <cit.>, Bishop and Jones <cit.>, and David and Semmes <cit.> on quantitative rectifiability of sets. The L^2 Jones function or geometric square function defined by 
    J_2(μ,x):=∫_0^1 β^h_2(μ,x,r)^2 dr/r
 records the total square error of the linear approximation numbers β^h_2(μ,x,r) on vanishing coarse scales.

Let (𝕏,ℳ)=(^n,ℬ_^n), let 𝒩 be the collection of rectifiable curves in ^n, and let ℱ:={^1 E:E is compact and ^1(E∩ B(x,r))∼_E r for all 0<r≤ E} denote the collection of  Hausdorff measures restricted to compact 1-dimensional Ahlfors regular sets. Pajot <cit.> proved that if μ=^1 E∈ℱ, then 
    μ^1_≡μ_𝒩=^1{x∈ E: J_2(μ,x)<∞},


    μ^1_≡μ_𝒩^⊥=^1{x∈ E: J_2(μ,x)=∞}.
 Pajot also obtained analogous results for m-dimensional Hausdorff measures restricted to compact m-dimensional Ahlfors regular sets using higher-dimensional beta numbers, which are defined using approximation by m-planes instead of approximation by lines.

Although the definition of J_2(μ,x) makes sense for any Radon measure, it  does not allow one to identify 1-rectifiable measures in general.
In <cit.>, Lerman explored modifications to the definition of β^h_2(μ,x,r) and J_2(μ,x) that expanded the class of measures to which Pajot's theorem applies. Further work in this direction (see <cit.>) led to a solution of the identification problem for 1-rectifiable Radon measures, which we will describe in detail in the next section. In the special case when μ is a pointwise doubling Radon measure (i.e. lim sup_r↓ 0μ(B(x,2r))/μ(B(x,r))<∞ at μ-a.e. x∈^n), the method of <cit.> can be used to show that the density-normalized L^2 Jones function 
    J_2(μ,x) := ∫_0^1 β^h_2(μ,x,r)^2·[r/μ(B(x,r))]^2 dr/r
 determines the 1-rectifiable and purely 1-unrectifiable part of μ: 
    μ^1_≡μ_𝒩=μ{x∈^n: J_2(μ,x)<∞},


    μ^1_≡μ_𝒩^⊥=μ{x∈^n: J_2(μ,x)=∞}.
 For a 1-dimensional Ahlfors regular measure, it is clear that J_2(μ,x)∼J_2(μ,x). Therefore, Badger and Schul's pointwise doubling theorem directly generalizes the case m=1 of Pajot's theorem to a wider class of measures, including measures like μ_0 from Example <ref> that are mutually singular with ^1. Martikainen and Orponen <cit.> have constructed a family of pointwise non-doubling Borel probability measures μ_ε on ^2 such that 
    _2(μ_ε,x)≤ε≪ 1  for all x∈^2
 and μ_ε is purely 1-unrectifiable. Thus, for general Radon measures, pointwise control on the density-normalized Jones function _2(μ,x) is also not enough to identify μ^1_.


Let (𝕏,ℳ)=(𝕏,ℬ_𝕏) be a connected metric space equipped with its Borel σ-algebra, let 𝒩 denote the collection of rectifiable curves in 𝕏, and let ℱ denote the collection of Radon measures on 𝕏 whose supports are 𝕏 (i.e.  μ(B)>0 for every ball B in 𝕏) and are doubling in the sense that 
    sup{μ(B(x,2r))/μ(B(x,r)):x∈𝕏, r>0}<∞.
 Azzam and Mourgoglou <cit.> proved that 
    μ^1_≡μ_𝒩=μ{x∈𝕏: lim inf_r↓ 0μ(B(x,r))/2r>0},
 
    μ^1_≡μ_𝒩^⊥=μ{x∈𝕏: lim inf_r↓ 0μ(B(x,r))/2r=0}.
  Standard examples (self-similar Cantor sets in ^2 of Hausdorff dimension one) show that the connectedness hypothesis cannot be dropped from Azzam and Mourgolgou's theorem. For examples in ^n of 1-rectifiable doubling measures whose support is the whole of ^n, see Garnett, Killip, and Schul <cit.>.

The examples above illustrate different solutions of Problem <ref> when 𝒩 is the collection of rectifiable curves in ^n and ℱ is one of several sets of σ-finite Borel measures on ^n. Additional results are available when 𝒩 is the collection of images of Lipschitz maps f:[0,1]^m→^n and μ≪^m (see 3), or when =ℍ^n is the n-th Heisenberg group (see 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 <ref>).

 Let 𝒩 denote the set of lines (1-dimensional affine subspaces) in ^2. Identify the Radon measures on ^2 that are carried by 𝒩 or singular to 𝒩.

The rest of this survey is organized, as follows. In 2, we present the solution of the identification problem for 1-rectifiable Radon measures from <cit.>. In 3, we review the current state of affairs on identification problems for m-rectifiable measures when m≥ 2.
In 4, we discuss further directions and open problems on generalized rectifiability, including fractional rectifiability and higher order rectifiability in ^n—and other spaces.






§ SOLUTION OF THE IDENTIFICATION PROBLEM FOR 1-RECTIFIABLE MEASURES


In this section, we will present the solution to the identification problem for 1-rectifiable Radon measures from <cit.> (see Theorem <ref>). The basic strategy is to promote a characterization of subsets of rectifiable curves, called the Analyst's Traveling Salesman theorem, to a characterization of 1-rectifiable measures. For any E⊂^n and bounded set Q⊂^n of positive diameter (such as a ball or a cube), the Jones' beta number β_E(Q) is the quantity in [0,1] defined by 
    β_E(Q):=inf_ℓsup_x∈ E∩ Q(x,ℓ)/ Q,
 where ℓ ranges over all straight lines in ^n, if E∩ Q≠∅, and by β_E(Q):=0 if E∩ Q=∅. At one extreme, if β_E(Q)=0, then E∩ Q is a subset of some straight line ℓ passing through Q. At the other extreme, if β_E(Q)∼ 1, then E∩ Q is “far away" from being contained in any line passing through Q. The following theorem was first conceived and proved with by Jones <cit.> when n=2 and later extended to higher dimensions n≥ 3 by Okikiolu <cit.>. For further information, see the survey <cit.> by Schul as well as the subsequent developments in the Heisenberg group by Li and Schul <cit.> and in Laakso type spaces by David and Schul <cit.>.

  Let n≥ 2 and let E⊂^n be a bounded set. Then E is contained in a rectifiable curve in ^n if and only if 
    S_E:=∑_Q dyadicβ_E(3Q)^2 Q <∞,
 where the sum ranges over all dyadic cubes and 3Q denotes the concentric dilate of Q. Moreover, if S_E<∞, there exists a rectifiable curve Γ containing E such that 
    ^1(Γ)≲_n  E+ S_E.


The characterization of 1-rectifiable measures from <cit.> has two main components: 

  * the lower 1-dimensional Hausdorff density 1(μ,·) of a measure μ (a common notion in geometric measure theory); and,

  * a density-normalized Jones function J^*_2(μ,·) associated to “anisotropic" L^2 beta numbers β^*_2(μ,·) (which are the main innovation in <cit.>).
 (Below I attempt to provide an intuitive explanation of some ideas behind the definition of β^*_2(μ,x) and J_2^*(μ,x) that R. Schul and I had in mind when we wrote, but did not expressly describe in <cit.>.)



For any Radon measure μ on ^n and x∈^n, the lower 1-dimensional Hausdorff density of μ at x is the quantity 1(μ,x) in [0,∞] given by 
    1(μ,x):=lim inf_r↓ 0μ(B(x,r))/2r.
 The following lemma shows that to identify the 1-rectifiable part of μ, it suffices to focus on the set of points where the lower density is positive.

  Let μ be a Radon measure on ^n. Then 
    μ^1_≤μ{x: 1(μ,x)>0}   and   μ{x:1(μ,x)=0}≤μ^1_.
 That is, μ{x:1(μ,x)=0} is purely 1-unrectifiable, and if μ is 1-rectifiable, then 1(μ,x)>0 at μ-a.e. x∈^n. 

 This is an easy consequence of the relationship between pointwise control on the lower density 1(μ,·) along a set E and the 1-dimensional packing measure 𝒫^1 of E (see Taylor and Tricot <cit.>); the finiteness of the packing measure 𝒫^1 on bounded sets in ; and, the interaction of packing measures with Lipschitz maps. See <cit.> for details.

For any Radon measure μ on ^n, bounded Borel set Q⊂^n with positive diameter (typically we take Q to be a ball or a cube), and straight line ℓ on ^n, we define the quantity β_2(μ,Q,ℓ) by 
    β_2(μ,Q,ℓ)^2=∫_Q ((y,ℓ)/ Q)^2 dμ(y)/μ(Q).
 The non-homogeneous 1-dimensional L^2 Jones beta number β_2(μ,Q) is given by 
    β_2(μ,Q):=inf_ℓβ_2(μ,Q,ℓ),
 where the infimum runs over all straight lines in ^n.

 When Q=B(x,r), we have 
    β^h_2(μ,x,r)^2= 4 μ(B(x,r))/r·β_2(μ,B(x,r))^2.
 Thus, the two variants of beta numbers are comparable at scales where μ(B(x,r))∼ r. One advantage of β_2(μ,B(x,r)) over β_2^h(μ,x,r) is that the former takes values in [0,1] and when μ=ℒ^n is the Lebesgue measure (for example), we have  
    β_2(ℒ^n,B(x,r))∼ 1  for all r>0,
 while the latter takes values in [0,∞) and when μ=ℒ^n we have  
    β_2^h(ℒ^n,x,r)∼ r^(n-1)/2→ 0  as  r→ 0.
 We interpret this to mean that for general Radon measures, for which there is no a priori control on the coarse density ratios μ(B(x,r))/r, the beta numbers β_2(μ,B(x,r)) with non-homogeneous scaling do better at indicating at a glance how well μ B(x,r) can be approximated by a measure supported on a line. The main reason that the numbers β_2^h(μ,x,r) persist in the literature is the historical accident of being initially defined that way by David and Semmes, who made a comprehensive investigation  into boundedness of singular integrals, where it was natural to restrict attention to Ahlfors regular measures (see <cit.>).

Before we define the anisotropic beta number β^*_2(μ,Q) from <cit.>, we give two lemmas in order to motivate its definition. The following observation, an elegant application of Jensen's inequality, is due to Lerman <cit.>. Practically speaking, it allows one to control the distance from the μ center of mass of a window Q to a straight line ℓ in terms of the quantity β_2(μ,Q,ℓ).

  Let μ be a Radon measure on ^n, let Q⊂^n be a bounded Borel set of positive diameter such that μ(Q)>0, and let 
    z_Q:=∫_Q z dμ(z)/μ(Q)∈^n
 denote the center of mass of Q with respect to μ. For every straight line ℓ in ^n, 
    (z_Q,ℓ)≤β_2(μ,Q,ℓ) Q.


 For every affine subspace ℓ in ^n, the function (·,ℓ)^2 is convex. Thus, 
    (z_Q,ℓ)^2 = (∫_Q z dμ(z)/μ(Q))^2≤∫_Q (z,ℓ)^2 dμ(z)/μ(Q) = β_2(μ,Q,ℓ)^2( Q)^2
 by Jensen's inequality.

Given two windows R and Q with R⊂ Q, the approximation number β_2(μ,Q) controls the approximation number β_2(μ,R) if one has control on Q /  R and μ(Q) / μ(R).

 Let μ be a Radon measure on ^n, let R,Q⊂^n be a bounded Borel set of positive diameter such that R⊂ Q and μ(R)>0. For any straight line ℓ in ^n, 
    β_2(μ,R,ℓ) ≤(μ(Q)/μ(R))^1/2 Q/ R β_2(μ,Q,ℓ).



 This is immediate from monotonicity of the integral. 

Now suppose that a window Q contains subregions R_1,…, R_m with R_i ∼ Q. (For example, imagine that Q is a dyadic cube and R_1,…,R_m are its dyadic descendants through some fixed number of generations.) Let z_R_1,…,z_R_m denote the μ centers of mass of R_1,…,R_m, respectively. Roughly speaking, if μ is a doubling measure, then we have a uniform bound on μ(Q)/μ(R_i), and thus, by Lemmas <ref> and <ref>, 
    (z_R_i,ℓ_Q) ≤ 2 β_2(μ,Q)  Q  for all 1≤ i≤ m,
 where is ℓ_Q is any line such that β_2(μ,Q,ℓ_Q)≤ 2 β_2(μ,Q). In other words, from just the doubling of μ and the definition of β_2(μ,Q), we can find a line ℓ_Q such that β_2(μ,Q) controls the distance of z_R_1,…,z_R_m to ℓ_Q. For a general Radon measure μ on ^n, it is not possible to uniformly bound the ratios μ(Q)/μ(R_i), and we cannot hope to control the distance of the points z_R_1,…,z_R_m to a common line using the number β_2(μ,Q) alone. Thus, following <cit.>, we modify the definition of β_2(μ,Q) to impose the missing control:

Let Q⊂^n be a dyadic cube. We say that a dyadic cube R⊂^n is nearby Q and write R∈Δ^*(Q) if 

  * R= Q or R=1/2 Q; and,

  * the concentric dilate 3R is contained in the concentric dilate 1600√(n) Q.
 Given a Radon measure μ on ^n and a dyadic cube Q⊂^n, we define β_2^*(μ,Q) by 
    β_2^*(μ,Q)^2 := inf_ℓsup_R∈Δ^*(Q)β_2(μ,R,ℓ)^2 min{μ(3R)/ R,1}.
 Let us temporarily ignore the truncation weight min{μ(3R)/ R,1} and see what we have gained by using nearby cubes R in the definition of β_2^*(μ,Q). Here we are using a dyadic cube Q⊂^n as a convenient short hand to represent a location and scale in ^n. Each dyadic cube Q can be viewed as the center of a window 1600√(n)Q that contains the subregions 3R for all R∈Δ^*(Q). The beta number β^*(μ,Q) is “anisotropic" because one normalizes the defining integrals for the β_2(μ,3R,ℓ) independently within each subregion of the window 1600√(n)Q, whereas the beta number β_2(μ,1600√(n)Q) for the full window uses a single normalization. From the definition of β_2^*(μ,Q), we can find a line ℓ such that 
    β_2(μ,3R,ℓ_Q)min{(μ(3R)/ R)^1/2,1}≤ 2β_2^*(μ,Q)
 for all R∈Δ^*(Q). Thus, by Lemma <ref>, we can control the distance of z_3R to ℓ_Q for all subregions 3R of 1600√(n)Q corresponding to R∈Δ^*(Q) (such that μ(3R)/ R≳ 1) using β_2^*(μ,Q). That is, we get uniform control of multiple μ centers of mass to a common line using a single approximation number, without needing to assume that μ is doubling!

For every Radon measure μ on ^n, we now define the (dyadic) density-normalized Jones function J^*_2(μ,·):^n→[0,∞] associated to the numbers β^*(μ,Q)  by 
    J_2^*(μ,x) := ∑_Q dyadic Q≤ 1β_2^*(μ,Q)^2  Q/μ(Q)χ_Q(x)  for all x∈^n.
 Suppose that Q_0 is a dyadic cube of side length at most 1 and let N<∞. Integrating J_2^*(μ,x) on the set {x∈ Q_0: J_2^*(μ,x)≤ N} with respect to μ, one obtains 
    ∑_Q∈𝒯β_2^*(μ,Q)^2 Q≤ Nμ(Q_0)<∞,
 where 𝒯 is the tree of dyadic cubes Q⊂ Q_0 such that 
    Q∩{x∈ Q_0: J_2^*(μ,x)≤ N}≠∅.
 For each c>0, let 𝒯_c be a maximal subtree where μ(3Q)/ Q≥ c for all Q∈𝒯_c. The argument that we sketched in the previous paragraph lets us control the distance of the μ centers of mass of triples of nearby cubes in 𝒯_c in terms of the numbers β_2^*(μ,Q). A technical extension of Jones' original traveling salesman construction proved in <cit.> then lets us build a rectifiable curve Γ containing the leaves of the tree 𝒯_c. (The constant 1600√(n) and number of generations of dyadic cubes (2) appearing in the definition of nearby cubes is chosen so that there are sufficiently many data points to enter into the extended traveling salesman construction, see <cit.>.) Running this procedure over a suitable countable choice of parameters, we arrive at Badger and Schul's solution to the identification problem for 1-rectifiable Radon measures:

 Let μ be a Radon measure on ^n. Then 
    μ^1_ = μ{x∈^n:1(μ,x)>0 and J_2^*(μ,x)<∞},
 
    μ^1_ = μ{x∈^n : 1(μ,x)=0 or J_2^*(μ,x)=∞}.



While Theorem <ref> cannot be directly applied to σ-finite Borel measures that are not locally finite, we note that it can be used indirectly, as follows. Suppose that ν is an infinite, σ-finite Borel measure on ^n. Then there exists a countable measurable partition of ^n into disjoint Borel sets {X_i}_1^∞ such that 0<ν(X_i)<∞ for all i≥ 1. Suppose we can find such a partition. Then the measure 
    ρ:=∑_i=1^∞1/2^iν(X_i)ν X_i
 is a Borel probability measure such that ν≪ρ≪ν. Theorem <ref> identifies a Borel set A_ρ such that ρ^1_ = ρ A_ρ and ρ^1_=ρ(^n∖ A_ρ). Since ν≪ρ, it follows that 
    ν^1_=ν A_ρ  and  ν^1_ = ν (^n∖ A_ρ),
 as well (which we leave as a simple exercise for the reader). In this sense, because we can solve the identification problem for any finite Borel measure ρ on ^n, we can also solve the problem for any σ-finite Borel measure ν on ^n.

For every Radon measure μ on ^n and dyadic cube Q⊂^n, one can define a variant β^**_2(μ,Q) of β^*(μ,Q) without the truncation weight by 
    β_2^**(μ,Q) := inf_ℓsup_R∈Δ^*(Q)β_2(μ,3R,ℓ).
 Also define the associated density-normalized Jones function J^**_2(μ,·):^n→[0,∞] by 
    J_2^**(μ,x) := ∑_Q dyadic Q≤ 1β_2^**(μ,Q)^2  Q/μ(Q)χ_Q(x)  for all x∈^n.
 The comparisons β_2^*(μ,Q)≤β_2^**(μ,Q) and J_2^*(μ,x)≤ J_2^**(μ,x) hold for all μ, Q, and x. In <cit.>, the authors proved the following theorems, which also follow from the argument outlined above. (In fact, R. Schul and I discovered Theorem <ref> before Theorem <ref>.)

  Let μ be a finite Borel measure on ^n with bounded support. Then μ is carried by a rectifiable curve (i.e. there is a rectifiable curve Γ such that μ(^n∖Γ)=0) if and only if 
    S_2^**(μ):=∑_Q dyadicβ_2^**(μ,Q)^2 Q<∞.
 Moreover, if S_2^**(μ)<∞, then there is a rectifiable curve Γ in ^n carrying μ such that ^1(Γ)≲_n (μ) + S_2^**(μ)

  Let μ be a Radon measure on ^n. Then we have μ{x: J_2^**(μ,x)<∞} is 1-rectifiable. (Hence μ{x: J_2^**(μ,x)<∞}≤μ^1_.)


One difference between Theorem <ref> and Theorem <ref> is that the latter does not require a bound on the lower Hausdorff density of μ. It is currently an open problem to decide whether or not for every Radon measure μ on ^n, 
    J_2^**(μ,x)<∞  at μ^1_-a.e. x∈^n.
 In other words, it is currently unknown whether or not the sufficient condition for a measure to be 1-rectifiable in Theorem <ref> is also a necessary condition. To overcome this gap and obtain a characterization, R. Schul and I (motivated by our earlier work) inserted the truncation weight 
    min{μ(3R)/ R,1}
 into the definition of β^**(μ,Q) to obtain β^*(μ,Q). This was plausible from the point of view of retaining the sufficient condition for rectifiability established in Theorem <ref> by Lemma <ref>. With the addition of the truncation weight, one can then follow the proof of the main theorem in <cit.> (using the “necessary" half of Theorem <ref>) and show that 
    ∫_Γ J_2^*(μ,x) dμ(x)<∞
 for any rectifiable curve Γ in ^n, from which it follows that J_2^*(μ,x)<∞ at μ^1_-a.e. x. See <cit.> for details. The idea of discounting the linear approximation numbers at scales of small density also appears in the recent work  of Naber and Valtorta <cit.> on a measure-theoretic version of the Reifenberg algorithm.

 The resolution of the identification problem for 1-rectifiable Radon measures provides a template for attacking similar problems. To solve the identification problem for Radon measures carried by a family of sets 𝒩 in a space , there are three basic steps.
First, find a suitable characterization of subsets of the sets in 𝒩. Second, transform the result for sets into a characterization of doubling measures carried by 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 ^n that are carried by m-dimensional Lipschitz graphs.



§ RECENT PROGRESS ON HIGHER DIMENSIONAL RECTIFIABILITY


Throughout this section, we let m and n denote integers with 1≤ m≤ n-1. Depending on the context, one encounters three possible definitions of m-rectifiable measures in ^n, which coincide for m-dimensional Hausdorff measures on m-sets, but differ for locally finite measures in general.  Let μ be a Radon measure on ^n. In decreasing order of generality, we say that (using the terminology in Definition <ref>) 

  * μ is (Lipschitz) image m-rectifiable if μ is carried by images of Lipschitz maps f:[0,1]^m→^n;

  * μ is (Lipschitz) graph m-rectifiable if μ is carried by isometric copies of graphs of Lipschitz maps g:^m→^n-m;

  * μ is C^1 m-rectifiable if μ is carried by m-dimensional embedded C^1 submanifolds of ^n.
 That is, (3)⇒(2)⇒(1) for every Radon measure μ, and moreover, both implications are trivial.
For restrictions of m-dimensional Hausdorff measures to m-sets and absolutely continuous measures, the three notions of m-dimensional rectifiability are equivalent:

 Suppose E⊂^n is ℋ^m measurable and 0<^m(E)<∞. If μ=^m E is image or graph m-rectifiable, then μ is C^1 m-rectifiable. More generally, if μ is a Radon measure on ^n and μ≪^m, then μ is m-rectifiable with respect to one of the definitions (1), (2), or (3) if and only if μ is m-rectifiable with respect to each of (1), (2), and (3).

 For the first part, see Federer <cit.>. The second part follows from the first with the aid of the Radon-Nikodym theorem.

In <cit.>, Martín and Mattila constructed, for all 0<s<1, sets E_s,F_s⊂^2 with 
    0<^s(E_s)<∞  and  0<^s(F_s)<∞
  such that ^s E_s is image 1-rectifiable, but not graph 1-rectifiable, and ^s F_s is graph 1-rectifiable, but not C^1 1-rectifiable. Thus, when working with general Radon measures, which are not necessarily absolutely continuous with respect to ^m, it is important to distinguish between image, graph, and C^1 rectifiability.

 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 μ on ^n is m-rectifiable if μ is image m-rectifiable. There are three basic reasons. First, image rectifiability is the definition that is consistent with the convention that a 1-rectifiable measure is a  measure carried by rectifiable curves, which is used implicitly and explicitly in the work of Besicovitch <cit.> and Morse and Randolph <cit.>. Second, in the monograph <cit.> that gave the field of geometric measure theory its name, Federer makes the definition that E⊂^n is countably (μ,m) rectifiable if μ E is image m-rectifiable 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 <cit.>. 

For any Radon measure μ on ^n and integer 1≤ m≤ n-1, let μ=μ^m_+μ^m_ denote the decomposition from Proposition <ref>, where μ^m_ is carried by images of Lipschitz maps f:[0,1]^m→ and μ^m_ is singular to images Lipschitz maps f:[0,1]^m→^n in the sense of Definition <ref>. The following fundamental problem in geometric measure theory about the structure of measures in Euclidean space is wide open.

[identification problem for m-rectifiable measures]  Let 2≤ m≤ n-1. Find geometric or measure-theoretic properties that identify μ^m_ and μ^m_ for every Radon measure μ on ^n. (Do not assume that μ≪^m.) 

Given a Radon measure μ on ^n and s>0, the lower and upper Hausdorff s-densities of μ are defined by 
    s(μ,x):=lim inf_r↓ 0μ(B(x,r))/ω_s r^s  and  s:=lim sup_r↓ 0μ(B(x,r))/ω_s r^s
 for all x∈^n, where ω_s is a constant depending on the normalization used in the definition of the s-dimensional Hausdorff measure. In particular, ω_s=2^s when  ℋ^s is defined by 
    ^s(E)=lim_δ↓ 0inf{∑_i=1^∞ ( E_i)^s: E⊂⋃_i=1^∞ E_i and  E_i≤δ}.

As with the case m=1, rectifiability of a measure restricts the appropriate lower density.

  Let μ be a Radon measure on ^n. Then 
    μ^m_≤μ{x: m(μ,x)>0}   and   μ{x:m(μ,x)=0}≤μ^m_.
 That is, μ{x:m(μ,x)=0} is purely m-unrectifiable, and if μ is m-rectifiable, then m(μ,x)>0 at μ-a.e. x∈^n. 

For absolutely continuous measures,  which are the Radon measures such that μ≪^m, or equivalently (see the exercises in <cit.>), the Radon measures such that 
    m(μ,x)=lim sup_r↓ 0μ(B(x,r))/ω_m r^m<∞  at μ-a.e. x∈^n,
 several solutions to Problem <ref> are available. At a high level, one of the reasons that Problem <ref> is easier for absolutely continuous measures is that for such measures the notions of Lipschitz image, Lipschitz graph, and C^1 rectifiability coincide. The following is the combined effort of several mathematicians (see below for a detailed citation).

 Let 1≤ m≤ n-1 be integers and let μ be a Radon measure on ^n. Assume μ≪^m (⇔ m(μ,x)<∞ at μ-a.e. x∈^n.) Then the following are equivalent: 

  * μ is m-rectifiable (carried by Lipschitz images of [0,1]^m);

  * there is a unique μ approximate tangent m-plane at μ-a.e. x∈^n;

  * μ is weakly m-linearly approximable at μ-a.e. x∈^n, i.e.  m(μ,x)>0 and 
    Tan(μ,x)⊆{cℋ^m L:c>0, L∈ G(n,m)}  at μ-a.e. x∈^n;


  * the m-dimensional Hausdorff density of μ exists and is positive μ-a.e.: 
    lim inf_r↓ 0μ(B(x,r))/ω_m r^m=lim sup_r↓ 0μ(B(x,r))/ω_m r^m>0  at μ-a.e. x∈^n;


  * m(μ,x)>0 at μ-a.e. x∈^n and the coarse density ratios are asymptotically optimally doubling in the sense that 
    lim_r↓ 0μ(B(x,2r))/ω_m(2r)^m-μ(B(x,r))/ω_m r^m=0   at μ-a.e. x∈^n;


  * m(μ,x)>0 at μ-a.e. x∈^n and the Jones function associated to the m-dimensional homogeneous L^2 beta numbers β_2^m,h(μ,x,r) is finite μ-a.e.: 
    ∫_0^1 β_2^m,h(μ,x,r)^2dr/r<∞   at μ-a.e. x∈^n.




 First, let us explain the various bits of notation and terminology used to state the theorem. We say that an m-dimensional affine subspace L of ^n containing x∈^n is a μ approximate tangent m-plane at x if m(μ,x)>0 and 
    lim sup_r↓ 0μ(B(x,r)∖ X)/ω_m r^m=0
 for every cone of the form X={y: (y,L)≤  c (y,L^⊥)}, where c>0 and L^⊥ denotes the unique (n-m)-plane  containing x that is orthogonal to L. Following <cit.>, we say that a non-zero measure ν is a tangent measure of μ at x if there exist sequences r_i↓ 0 and 0<c_i<∞ such that the rescaled measures μ_i, defined by 
    μ_i(E)=c_iμ(x+r_iE),
 converge weakly to ν in the sense of Radon measures. We denote the set of all tangent measures of μ at x by Tan(μ,x). As usual, the set G(n,m) denotes the Grassmannian of m-dimensional linear subspaces of ^n. Following <cit.>, the homogeneous m-dimensional L^2 Jones beta number β_2^m,h(μ,x,r) is defined by 
    β_2^m,h(μ,x,r)^2:=inf_L ∫_B(x,r)((x,L)/r)^2dμ(x)/r^m,
 where the infimum runs over all m-dimensional affine subspaces of ^n.

The implications (1)⇒(2)⇒(4) essentially follow from Rademacher's theorem on almost everywhere differentiability of Lipschitz maps and may be considered elementary. The implication (2)⇒(1) was proved by Federer <cit.>. The implication (2)⇒(3) is trivial. The implication (3)⇒(1) was proved by Mattila <cit.> (extending the special case m=2 and n=3 by Marstrand <cit.>). The implication (4)⇒(3) was proved by Preiss <cit.>. Thus, the equivalence of (1), (2), (3), and (4) was finally completed after a span of 40 years from the 1940s to the 1980s. The essential difficulty in the final implication (4)⇒(3) is the (surprising!) existence of non-flat m-uniform measures in ^n, i.e. Radon measures in ^n such that 
    μ(B(x,r))=cr^m  for all x∈μ and r>0,
 for which the support of μ is not an m-plane. For further background and the most recent developments on the uniform measure classification problem, including new examples of 3-uniform measures in ^n, for each n≥ 5, see Nimer <cit.>. For a friendly presentation of the proof of (1)⇔(2)⇔(3)⇔(4),  see the monograph <cit.> by De Lellis.

The equivalence of (5) and (6) with (1), (2), (3), and (4) are recent developments. The implication (4)⇒(5) is trivial. The implication (5)⇒(4) was proved by Tolsa and Toro <cit.>. The implication (1)⇒(6) was proved by Tolsa <cit.>. With the additional assumption m(μ,x)>0 μ-a.e., the implication (6)⇒(1) was proved by Pajot <cit.> (for Radon measures of the form μ=^m E, E compact) and Badger and Schul <cit.>. With the a priori weaker assumption m(μ,x)>0 μ-a.e., the implication (6)⇒(1) was proved by Azzam and Tolsa <cit.>. To obtain the last result, Azzam and Tolsa carry out an intricate stopping time argument using David and Mattila's “dyadic" cubes <cit.> and David and Toro's extension of Reifenberg's algorithm to sets with holes <cit.>.


For related work on rectifiability of  absolutely continuous measures and the theory of mass transport, see Tolsa <cit.>, Azzam, David, and Toro <cit.>, and Azzam, Tolsa, and Toro <cit.>. For the connection between rectifability of sets and Menger-type curvatures, see  Léger <cit.>, Lerman and Whitehouse <cit.>, Meurer <cit.>, and Goering <cit.>. For related results about discrete approximation and rectifiability of varifolds, see Buet <cit.>.


In recent work <cit.>, Edelen, Naber, and Valtorta provide new sufficient conditions for qualitative and quantitative rectifiability of Radon measures μ on ^n that do not require absolute continuity, μ≪^m (⇔m(μ,x)<∞ at μ-a.e. x∈^n.) For simplicity, we state a special case of their main result.

  Let μ be a Radon measure on ^n. Assume that m(μ,x)<∞ and m(μ,x)>0 at μ-a.e. x∈^n. Then 
    ∫_0^1 β_2^m,h(μ,x,r)^2dr/r<∞   at μ-a.e. x∈^n
 implies μ is m-rectifiable (and hence m(μ,x)>0 μ-a.e.)

 Edelen, Naber, and Valtorta's proof of Theorem <ref> is based on an updated, quantitative version of the Reifenberg algorithm (cf. <cit.> and <cit.>), 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 <ref> has been provided by Tolsa <cit.> using a method closer to that of <cit.>. It is evident that the sufficient condition for a measure to be m-rectifiable in Theorem <ref> is not necessary in view of the example of Martikainen and Orponen <cit.> from the case m=1. 

At the root, the main challenge in solving Problem <ref> is the lack of a good Lipschitz parameterization theorem for sets of dimension m≥ 2. If f:[0,1]^m→^n is Lipschitz, then Γ=f([0,1]^m) is compact, connected, locally connected, and ℋ^m(Γ)<∞. When m=1, the converse of this fact is also true. In fact, it has been known for over 90 years that if Γ⊂^n is a nonempty continuum with ^1(Γ)<∞, then Γ is a Lipschitz image of [0,1] (see <cit.> where the theorem is attributed to a paper of Ważewski from 1927, and for a simple proof, see the appendix of <cit.>.) The situation in higher dimensions is quite different. For example, let C⊂[0,1]^2 be any self-similar Cantor set of Hausdorff dimension 1 (so that ^1 C is purely 1-unrectifiable by Hutchinson <cit.>.) Then ^2 (C×[0,1]) is purely 2-unrectifiable by Theorem <ref>(2). By adjoining a square to the base of C×[0,1], we obtain a set K, 
    K:=(C×[0,1])∪([0,1]^2×{0})⊂^3,
 which is compact, connected, path-connected, and Ahlfors 2-regular (hence ^2(Γ)<∞), but K is not locally connected, and therefore, K is not a Lipschitz image of [0,1]^2 nor a continuous image of [0,1]. From this one sees that being a continuum with finite or Ahlfors regular ^2 measure does not ensure a set is locally connected. In fact, it turns out that even with an added assumption of local connectedness, one still cannot guarantee the existence of a Lipschitz parameterization for Ahlfors regular contiuum when m≥ 2. The following result will appear in forthcoming work by the author, Naples, and Vellis.

 For all m≥ 2, there exist compact, connected, locally connected, m-Ahlfors regular sets G⊂^n such that G is not contained in the image of any Lipschitz map f:[0,1]^m→^m+1.

 Start with the set K above. One can transform K into a locally connected set G by adjoining sufficiently many squares S_j,k×{t_k}⊂^2×, 1≤ j≤ J_k on a countable dense set of heights t_k∈ [0,1], with diameters of S_j,k vanishing as k→∞. By carefully selecting parameters, one can ensure that G is Ahlfors 2-regular. However, since G contains C×[0,1] and ^2(C×[0,1]) is purely 2-unrectifiable, it is not possible that G is contained in the image of a Lipschitz map f:[0,1]^2→^3.

In order to attack Problem <ref>, 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).

  For each m≥ 2, find sufficient geometric, metric, and/or topological conditions that ensure a set Γ⊂^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 <ref>. One promising alternative is a class of surfaces that support a traveling salesman type theorem, recently identified by Azzam and Schul <cit.>, which are lower regular with respect to the m-dimensional Hausdorff content ^m_∞(E):=inf{∑ ( E_i)^m:E⊂⋃_i E_i}. For a complete description, we refer the reader to <cit.>. Also see Villa <cit.>, which characterizes the existence of approximate tangent m-planes of content lower regular sets.








§ FRACTIONAL RECTIFIABILITY AND OTHER FRONTIERS




 §.§ Fractional rectifiability


Let 1≤ m≤ n-1 be integers and let s be a real number with s≥ m. Recall that a map f:[0,1]^m→^n is (m/s)-Hölder continuous if there is a constant H<∞ such that 
    |f(x)-f(y)|≤ H|x-y|^m/s  for all x,y∈[0,1]^m.
 Alternatively, one may view the map f as a Lipschitz map from the snowflaked metric space ([0,1]^m,|·|^m/s) to ^n. Below we call the image Γ=f([0,1]^m) of such a map a (m/s)-Hölder m-cube, or when m=1, we may call Γ a (1/s)-Hölder curve.
From the definition of Hausdorff measures, it easily follows that 
    ^s(Γ)≲_m,s,H^m ([0,1]^m)<∞.
 Thus, every (m/s)-Hölder m-cube Γ⊂^n is a compact, connected, locally connected set with ^s(Γ)<∞. Martín and Mattila <cit.> suggested using images of (m/s)-Hölder maps from ^m as a basis for studying “fractional rectifiability" of ^s measures restricted to s-sets E⊂^n, i.e.  ^s measurable sets such that 0<^s(E)<∞. In <cit.>, V. Vellis and I revisited Martín and Mattila's idea in the context of arbitrary Radon measures:

By Proposition <ref>, any Radon measure μ on ^n can be decomposed as 
    μ=μ_m→ s+μ_m→ s^⊥,
 where μ_m→ s is carried by (m/s)-Hölder m-cubes and μ_m→ s^⊥ is singular to (m/s)-Hölder m-cubes. When s=m, the measures 
    μ_m→ m=μ^m_  and  μ_m→ m^⊥=μ^m_
 are the m-rectifiable and purely m-unrectifiable parts of μ, respectively.
The existence of (m/n)-Hölder surjections f:[0,1]^m→[0,1]^n, commonly called space-filling maps (see <cit.> for the case m=1 and <cit.>, <cit.> for m≥ 2),  guarantees that μ_m→ s=μ and μ_m→ s^⊥=0 for all s≥ n. Also, Theorem <ref> above identifies μ_1→ 1 and μ_1→ 1^⊥ for all Radon measures. All other cases of the following problem are open.

[identification problem for fractional rectifiability]  Let 1≤ m≤ n-1 be integers and let s∈[m,n). Find geometric or measure-theoretic properties that identify μ_m→ s and μ_m→ s^⊥ for every Radon measure μ on ^n. (Do not assume that μ≪^s.) 

Hausdorff measures on self-similar sets provide essential examples of rectifiable and purely unrectifiable behavior in fractional dimensions. For further results in this direction, see Martín and Mattila <cit.> and Rao and Zhang <cit.>.

  Let 1≤ m≤ n-1 and let S=⋃_i=1^k f_i(S)⊂^n be a self-similar set of Hausdorff dimension s∈[m,n) in the sense of Hutchinson <cit.>. Assume that 
    f_i(S)∩ f_j(S)=∅  for all i≠ j,
 Then ^s S is singular to (m/s)-Hölder m-cubes.


 Let S=⋃_1^kf_i(S)⊂^n be a self-similar set of Hausdorff dimension s∈[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 m-cubes.

  Let 1≤ m≤ n-1 be integers, let s∈[m,n), and let t∈[0,s). If μ is a Radon measure on ^n, then 

  * μ{x:s(μ,x)=0} is singular to  (m/s)-Hölder m-cubes;

  * ν is carried by (1/s)-Hölder curves, where 
    ν≡μ{x:∫_0^1 r^s/μ(B(x,r))dr/r<∞ and lim sup_r↓ 0μ(B(x,2r))/μ(B(x,r))<∞};


  * ρ≡μ{x:0<t(μ,x)≤t(μ,x)<∞} is carried by (m/s)-Hölder m-cubes;

  * moreover, if 0≤ t<1, then ρ is carried by bi-Lipschitz embeddings f:[0,1]→^n.



[2^n-corner Cantor Sets] Let E_t⊂^n be a 2^n-corner Cantor set of Hausdorff dimension 0<t<n, generated starting from a cube of side length 1 by replacing it with 2^n subcubes of side length 2^-n/t, placed at the corners of the original cube (see Figure <ref> for a depiction of the case n=3). Then ^t E_t is Ahlfors t-regular, i.e.  ^t(E_t∩ B(x,r))∼ r^t for all x∈ E_t and 0<r≤ 1. Thus, for any integer 1≤ m≤ n-1 such that t≥ m, 

  * ^t E_t is singular to (m/t)-Hölder m-cubes by Theorem <ref>; and,

  * ^t E_t is carried by (m/s)-Hölder m-cubes for all s>t by Theorem <ref>(3). Moreover, if 0<t<1, then ^t E_t is carried by bi-Lipschitz curves by Theorem <ref>(4).




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 <ref>).

 For all 1<s<2, find sufficient geometric, metric, and/or topological conditions that ensure a set Γ⊂^n is (contained in) a (1/s)-Hölder curve.



 §.§ Higher-order rectifiability


Let 1≤ m≤ n-1 and k≥ 1 be integers, and let α∈[0,1]. We say that a Radon measure μ on ^n is C^k,α m-rectifiable if μ is carried by m-dimensional C^k,α embedded submanifolds of ^n. In the case α=0, we also say that μ is C^k m-rectifiable. The study of higher-order rectifiability of measures was initiated by Anzellotti and Serapioni <cit.>. In general, different orders of rectifiability give rise to different classes of measures.

 Let 1≤ m≤ n-1 and k,l≥ 1 be integers, and let α,β∈[0,1]. 

  * If k+α < l+β, then there exists an m-set E⊂^n such that ^m E is C^k,α m-rectifiable and purely C^l,β m-unrectifiable (i.e. singular to C^l,β submanifolds).

  * If μ is a Radon measure on ^n and μ≪^m, then μ is C^k,1 rectifiable if and only if μ is C^k+1 rectifiable.


 For (1), see <cit.>. When μ=^m E and 0<^m(E)<∞, (2) is a consequence of <cit.>. The assertion for absolutely continuous measures then follows via the Radon-Nikodym theorem. Compare to Theorem <ref> above. 

Assume that M⊂^n with ^m(M)<∞. Let x∈ M and assume that M admits a unique approximate tangent plane T_x M at x. Following <cit.>, we say that a closed set Q_x M⊂^n is the approximate tangent paraboloid of M at x if 

  * Q_xM={(y,A(y,y)):y∈ T_x M} for some bilinear symmetric form 
    A:T_xM × T_x M → (T_xM)^⊥,


  * ^mξ_x,ρ(M) converges to ^m Q_x M weakly in the sense of Radon measures as ρ↓ 0, where ξ_x,ρ denotes the nonhomogeneous dilation of M at x defined by 
    ξ_x,ρ:^n→^n,  ξ_x,ρ(y)= ρ^-1π_T_xM(y-x) + ρ^-2π_T_xM^⊥(y-x).

    

[approximate tangent paraboloid to a graph] Let f:^m→^n-m be a function of class C^2 such that f(0)=0, Df(0)=0, and 
    f(x)=1/2∑_p,q=1^n D_p,q^2f(0)x_px_q + o(|x|^2)=:g(x)+o(|x|^2).
 Then the graph of f is a C^2 manifold and Q_0(graph f) = {(x,g(x)):x∈^m}.

Anzellotti and Serapioni <cit.> give geometric characterizations of C^1,α rectifiability when α∈[0,1]. In the case α=1, their result is the following.

 Assume M⊂^n with ^m(M)<∞. Then ^m M is C^2 m-rectifiable if and only if ^m(M∖⋃_j=1^∞ M_j)=0 for some M_j⊂^n with M_i∩ M_j=∅ for all i≠ j and such that for every j≥ 1 and ^m-a.e. x∈ M_j, 

  * the approximate tangent plane T_x M_j exists,

  * the approximate tangent paraboloid Q_x M_j exists, and

  * (see <cit.> for definition of approximate limits) 
    ap lim sup_y→ xy∈ M_jd(T_yM_j,T_xM_j)|y-x|<∞,   d(T_1,T_2)≡sup_|y|=1 |π_T_1(y)-π_T_2(y)|.



More recently, two new characterizations of C^1,α rectifiability of absolutely continuous measures have been provided by Kolasiński <cit.> (using Menger-type curvatures) and by Ghinassi <cit.> (using L^2 beta numbers), the latter of which can be stated as follows. See (<ref>) for the definition of β_2^m,h(μ,x,r).

 Let μ be a Radon measure on ^n with μ≪^m. If m(μ,x)>0 at μ-a.e. x∈^n and 
    ∫_0^1 β_2^m,h(μ,x,r)^2/r^2αdr/r<∞  at μ-a.e. x∈^n,
 then μ is C^1,α m-rectifiable.

It would be interesting to know whether Ghinassi's sufficient condition is also necessary for C^1,α rectifiability (compare to Theorem <ref>(6)). The following problem is also open.

[identification problem for C^k,α m-rectifiable measures]  Find geometric or measure-theoretic properties that characterize C^k,α m-rectifiable measures when k≥ 2. (To start, you may assume μ≪^m.) 

For recent progress on Problem <ref> for Hausdorff measures, see Santilli <cit.>.




 §.§ Rectifiability in other spaces
 Following Kirchheim <cit.>, a metric space (X,ρ) is called m-rectifiable if the m-dimensional Hausdorff measure on X is carried by images of Lipschitz maps from subsets of ^m into X. When E⊂^n, the space (E,d_2) equipped with induced Euclidean metric is m-rectifiable if and only if ^m E is Lipschitz image rectifiable in the sense of 3. Kirchheim examined the structure of general m-rectifiable metric spaces and proved that for those spaces with ^m(X)<∞, the m-dimensional Hausdorff density exists and equals 1 at ^m almost every point.




  Let (X,ρ) be a metric space and assume that ^m(X)<∞ for some integer m≥ 1. If X is m-rectifiable, then 
    lim_r↓ 0^m(B_ρ(x,r))/ω_m r^m=1  at ^m-a.e. x∈ X.


When (X,ρ)=(E,d_2), where E⊂^n is equipped with the Euclidean metric and ^m(E)<∞, the converse of Theorem <ref> is also true. This was proved by Besicovitch <cit.> when m=1 and n=2 and by Mattila <cit.> for general m and n. The converse of Theorem <ref> is also true when m=1 for general metric spaces with ^1(X)<∞  by Preiss and Tišer <cit.> (recall Example <ref> above). In all other cases it is not presently known whether the converse of Kirchheim's theorem is true or false.

 Let m≥ 2. Prove that for every metric space (X,ρ) with ^m(X)<∞ that (<ref>) implies X is m-rectifiable. Or find a counterexample.

Problem <ref> is interesting even in the case when (X,ρ)=(E,d_p) for some E⊂^n and d_p is the distance induced by the p-norm, p≠ 2. For related work on existence of densities of measures in Euclidean spaces with respect to non-spherical norms, see the series of papers by Lorent <cit.>.

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 <cit.>. Lipschitz differentiability spaces were introduced by Cheeger <cit.> and examined in depth by Bate <cit.>. 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 <cit.> for a detailed description and several characterizations of differentiability spaces. The following theorem is a simplified statement of Bate and Li's main result.

 A metric space (X,ρ) is m-rectifiable if and only if ^m X is carried by Borel sets U⊂ X such that 

  * (U,ρ,^m) is an m-dimensional Lipschitz differentiability space, and 
  * 0<m(^m U,x)≤m(^m U,x)<∞ at ^m-a.e. x∈ U.

A metric space (X,ρ) is called purely m-rectifiable if ^m X is singular to images of Lipschitz maps from subsets of ^m into X. For example, the (first) Heisenberg group ℍ with topological dimension 3 and Hausdorff dimension 4 is purely m-unrectifiable for all m=2,3,4 (see Ambrosio and Kirchheim <cit.>). 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, Serapioni, and Serra Cassano <cit.>. For related developments, see <cit.> and <cit.>.
A characterization of complete, purely m-unrectifiable metric spaces with ^m(X)<∞ was recently announced by Bate <cit.>. A related sufficient condition for rectifiability of a metric space (using Bate's theorem) has been announced by David and Le Donne <cit.>.


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