Rectifiability of pointwise doubling measures in Hilbert Space
aa r X i v : . [ m a t h . C A ] F e b RECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE
LISA NAPLESA
BSTRACT . In geometric measure theory, there is interest in studying the interaction of measureswith rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to char-acterize rectifiable pointwise doubling measures in Hilbert space. Given a measure µ , we constructa multiresolution family C µ of windows, and then we use a weighted Jones’ function ˆ J ( µ, x ) torecord how well lines approximate the distribution of mass in each window. We show that when µ is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the con-verse also holds. Additionally, we present an algorithm for the construction of a rectifiable curveusing appropriately chosen δ -nets. Throughout, we discuss how to overcome the fact that in infinitedimensional Hilbert space there may be infinitely many δ -separated points, even in a bounded set.Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs. C ONTENTS
1. Introduction 11.1. Background 11.2. Preliminaries 41.3. Outline 71.4. Acknowledgment 72. Necessary condition for rectifiability 83. Sufficient condition for rectifiability 144. Proof of Theorem B 215. An example of pointwise doubling measure with infinite dimensional support 226. Drawing curves through nets: an Analyst’s Traveling Salesman Algorithm 236.1. Description of curves 246.2. Connectedness 276.3. Length estimates 287. Graph rectifiable measures 34Appendix A. 38References 401. I
NTRODUCTION
Background.
One goal of geometric measure theory is to understand the global structure ofa measure through analysis of local geometric data. We use the below terminology to formalizethis notion.
Date : February 18, 2020.2010
Mathematics Subject Classification.
Primary 28A75.
Key words and phrases. -rectifible measures, purely -unrectifiable measures, doubling measures, Analyst’s trav-eling salesman theorem, Jones’ beta numbers. Definition 1.1.
Let ( X , M ) be a measurable space, and let N ⊂ M be a family of measurablesets. We say (1) µ is carried by N if there exist countably many N i ∈ N such that µ ( X \ S i N i ) = 0 ; (2) µ is singular to N if µ ( N ) = 0 for every N ∈ N . A σ -finite measure µ on ( X , M ) can be decomposed uniquely as µ = µ N + µ ⊥N where µ N is carried by N and µ ⊥N is singular to N . In [Bad19], Badger poses the followingproblem: Problem 1 (Identification Problem) . Let ( X , M ) be a measurable space, let N ⊂ M be a familyof 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 { x ∈ X : Q ( µ, x ) holds } . That is, we seek to find pointwise properties P ( µ, x ) and Q ( µ, x ) that identify the part of µ wherethe underlying geometric structure agrees with the structure of sets in N and the part of µ wherethe underlying geometric structure is distinct from that of the sets in N . There is particular interestin understanding the conditions under which measures can be decomposed when X is a metricspace, M contains the Borel sets, and N is the collection of rectifiable curves , that is, compact,connected sets of finite length. Measures µ which are carried by the collection of rectifiable curvesare called rectifiable measures , and measures which are singular to the collection of rectifiablecurves are called purely unrectifiable measures . We use the notation(1) µ = µ rect + µ pu to indicate decomposition of the measure µ into a rectifiable component and a purely unrectifiablecomponent. We remark that the class of rectifiable curves agrees with the class of images of theunit interval under Lipschitz maps, f ([0 , , where f : [0 , → X is Lipschitz. Therefore, in ourdiscussion of rectifiable and purely unrectifiable measures we will freely move between discussingcompact, connected sets of finite length and images of Lipschitz maps.The study of rectifiable measures stems from the study of rectifiable sets. A rectifiable set is aset which is contained H -a.e. in a countable union of rectifiable curves, where H denotes the1-dimensional Hausdorff measure. For an introduction to Hausdorff measures, see e.g. [Mat95,Section 4.3]. Given an arbitrary set in R n of finite length, we cannot expect the set to admit tangentlines at typical points. However, by Rademacher’s Theorem, a Lipschitz map f : [0 , → R n isdifferentiable L -a.e., and thus at H -a.e. x ∈ f ([0 , there is a unique tangent given by thederivative map. A rectifiable set can inherit the tangents from the rectifiable curve in which it iscontained. The notion of rectifiable sets in the plane was originally introduced by Besicovitch[Bes28]. Morse and Randolph [MR44] and Federer [Fed47] extended the concept of rectifiablesets to measures in Euclidean space. Since then a large theory has been developed for identifyingrectifiable measures (and their higher-dimensional analogues) µ under the additional assumptionof absolute continuity of µ with respect to 1-dimensional Hausdorff measure ( µ ≪ H ). Imposingthe absolute continuity assumption on measures allows one to replace the class of Lipschitz imageswith the class of bi-Lipschitz images or Lipschitz graphs in the definition of rectifiable measure.For results in this direction see [Mat75], [Pre87], [AT15], [ATT18], [Dab19a] and [Dab19b]. How-ever, Garnett, Killip, and Schul [GKS10] constructed a doubling measure on R n which is both car-ried by Lipschitz images and singular to every bi-Lipschitz image. Thus they showed that the class ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 3 of Radon measures carried by bi-Lipschitz images is strictly smaller than the class of measurescarried by Lipschitz images. In what follows, we adopt Federer’s convention [Fed47], [Fed69] anddo not assume a priori that µ is absolutely continuous with respect to H .Badger and Schul [BS15], [BS17] characterized rectifiable Radon measures on R n in terms of L Jones’ beta numbers and a density adapted Jones function. Jones’ beta numbers for sets wereoriginally introduced by Peter Jones [Jon90] as a means to solve his Analyst’s Traveling SalesmanProblem that asked to give necessary and sufficient conditions for a set to be contained in a singlerectifiable curve. Jones provided a solution to his problem for sets in R and Okikiolu [Oki92]extended the result to R n . Later, Jones’ result was extended to Hilbert space by Schul [Sch07]. Wesummarize the result for Hilbert space here. Definition 1.2 (Beta number) . Let E ⊂ H , where H is a separable, infinite dimensional Hilbertspace or R n , and let Q ⊂ H be bounded. We define β E ( Q ) ∈ [0 , by inf ℓ sup x ∈ E ∩ Q dist( x, ℓ )diam Q , where ℓ ranges over all lines ℓ in H . If E ∩ Q = ∅ , we set β E ( Q ) = 0 . The beta numbers measure how well the set E is approximated by a line in the window Q .In Euclidean space, dyadic cubes are often a practical choice for windows. However, in infinitedimensional Hilbert space, cubes are no longer practical because each cube has infinite diameterand infinitely many children. To prove an Analyst’s Traveling Salesman theorem, Schul replaceddyadic cubes with a multiresolution family of balls G K for a set bounded K ⊂ H , defined asfollows. Fix k such that k ≥ diam( K ) . For each k ≥ k , let N Kk ⊃ N Kk − be a maximal − k -netfor K . Set U k,i := B ( n i , λ − k ) to be the closed ball of radius λ − k centered at n i ∈ N k ; wespecify λ > later. We denote the collection of balls arising from the nets N Kk by G Kk and we set G K = ∞ [ k = k G Kk . Unlike dyadic cubes which are intrinsic to Euclidean space, the multiresolution family depends onthe set K as well as on the specific choice of the net N Kk . Theorem 1.1 (See [Sch07], Theorem 1.1 and Theorem 1.5) . space (1) (Necessary Condition) Let Γ be any connected set containing K . Then X U ∈ G K β ( U ) diam( U ) . H (Γ) . The constant behind the symbol . depends only on the choice of λ . (2) (Sufficient Condition) There is a constant λ such that for all λ > λ and for any set K ⊂ H there exists a connected set Γ ⊃ K satisfying H (Γ ) . diam( K ) + X U ∈ G K β K ( U ) diam( U ) . Here we require − k ≥ diam( K ) . For the study of rectifiable measures, Jones’ beta numbers are replaced by an L variant whichweigh the distances of points from a line according to the mass distribution of µ . See e.g. [BS15],[BS17]. LISA NAPLES
Definition 1.3 ( L beta number) . Let µ be a locally finite Borel measure on H , a separable infinitedimensional Hilbert space or R n . Let E ⊂ H be a bounded subset. We define β ( µ, E ) by β ( µ, E ) = inf ℓ Z E (cid:18) dist( x, ℓ )diam E (cid:19) dµ ( x ) dµ ( E ) where the infimum is taken over all lines ℓ in H . In the case that µ ( E ) = 0 , we define β ( µ, E ) = 0 . The L beta number measures the concentration of mass near lines in a particular window E .To prove results about measures on Euclidean space R n , Badger and Schul recorded beta numberson the collection of half-open dyadic cubes of side length at most , ∆ ( R n ) , using the density-normalized L Jones function ˜ J ( µ, · ) defined by ˜ J ( µ, x ) := X Q ∈ ∆ ( R n ) β ( µ, Q ) diam Qµ ( Q ) χ Q ( x ) for all x ∈ R n . Similar to above, when µ ( Q ) = 0 , we interpret β ( µ, Q ) diam Q/µ ( Q ) = 0 .Although Badger and Schul proved results for general Radon measures, here we only state theirresult for pointwise doubling measures, which has lighter notation. Theorem 1.2 ([BS17], Theorem E) . Let n ≥ . If µ is a Radon measure on R n such that at µ -a.e. x lim sup r ↓ µ ( B ( x, r )) /µ ( B ( x, r )) < ∞ then the decomposition µ = µ rect + µ pu is given by (2) µ rect = { x ∈ R n : ˜ J ( µ, x ) < ∞} , (3) µ pu = { x ∈ R n : ˜ J ( µ, x ) = ∞} . Preliminaries.
In this paper, we extend the results of Badger and Schul to pointwise doublingmeasures on a separable infinite dimensional Hilbert space, H . Following [Sch07], we replace thedyadic cubes used in the Euclidean case with a multiresolution family of balls. However, weconstruct the multiresolution family with respect to a carrying set of µ . Fix some such set X ⊂ H so that µ ( H \ X ) = 0 . For example, we may choose X = spt ( µ ) , where spt ( µ ) is the largest closedsubset of H such that for all x in the subset, every neighborhood of x has positive measure. Thenfix an integer k . We denote a maximal − k -net of X by X µk . We choose the nets X µk to be nestedso that X µk +1 ⊃ X µk for all k ≥ k . For a net X µk , we define an associated collection of closed balls, C µk = (cid:8) B ( x jk , λ − k ) : x jk ∈ X µk (cid:9) , where λ > is some fixed constant. We will specify conditions on λ later in the exposition.Then we set C µ := ∞ [ k = k C µk , and we call C µ a multiresolution family of balls for the measure µ . We emphasize that the collec-tion C µ is dependent on the measure µ and more specifically on the choice of nets X µk . We use thenotation B jk := B ( x jk , λ − k ) , and for a fixed ball B ∈ C µ , we denote the center by x B . For c > , we define cB jk = B ( x jk , cλ − k ) . ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 5
That is, cB is the dilation of ball B by a factor of c .We define the L density adapted Jones function ˆ J ( µ, r, · ) on H to be ˆ J ( µ, r, x ) := X B ∈ C µ Radius ( B ) ≤ r β ( µ, B ) diam Bµ ( B ) χ B ( x ) for all x ∈ H . When r = 1 , we abbreviate ˆ J ( µ, x ) = ˆ J ( µ, , x ) . The following lemma will beuseful. Lemma 1.1 (cf. [BS15], Lemma 2.9) . For every locally finite Borel measure µ , the sets { x ∈ H : ˆ J ( µ, r, x ) < ∞} and { x ∈ H : ˆ J ( µ, r, x ) = ∞} are independent of the parameter r . Before we state our main result, we provide the following definition.
Definition 1.4 (Pointwise doubling measure) . We say a measure µ on H is pointwise doubling if µ is finite on bounded sets and for µ -a.e. x , lim sup r ↓ µ ( B ( x, r )) µ ( B ( x, r )) < ∞ . Furthermore, we say that µ is a doubling measure if there exists a constant D such that for all r > and µ -a.e. x , µ ( B ( x, r )) ≤ Dµ ( B ( x, r )) . Theorem A (Characterization of rectifiable doubling measures) . Let µ be a pointwise doublingmeasure on a separable, infinite dimensional Hilbert space H . Then µ is rectifiable if and only if ˆ J ( µ, x ) < ∞ for µ -a.e. x ∈ H. We will freely refer to the necessary condition and the sufficient condition of Theorem A.Necessary condition: If µ is rectifiable, then ˆ J ( µ, x ) < ∞ for µ -a.e. x ∈ H. Sufficient condition: If ˆ J ( µ, x ) < ∞ for µ -a.e. x, then µ is rectifiable. Theorem B (Decomposition theorem for doubling measures) . Let µ be a pointwise doubling mea-sure on a separable, infinite dimensional Hilbert space H . Then the decomposition µ = µ rect + µ pu is given by (4) µ rect = { x ∈ H : ˆ J ( µ, x ) < ∞} , (5) µ pu = { x ∈ H : ˆ J ( µ, x ) = ∞} . One of the challenges of proving the characterization results in infinite dimensional space asopposed to R n arises in the differences between the multiresolution family of balls and dyadiccubes. In particular, the set of dyadic cubes satisfies convenient counting properties. For a givenhalf-open dyadic cube Q ∈ R n of side length − k , there are n dyadic cubes of side length − ( k +1) contained in Q . Additionally, cQ intersects at most C ( c, n ) other cubes of side length − k where C ( c, n ) is a constant which depends only on the dilation constant c and the dimension of the space n . The pointwise doubling condition assumed on µ allows us to recover some of the countingproperties of dyadic cubes for subcollections of C µ . We say a subcollection C ′ ⊂ C µ satisfies the finite overlap condition with respect to µ if there exist constants P µj − k = P ( C ′ , j − k ) , j ≥ k , suchthat for any ball B = B ( x, λ − k ) ∈ C ′ , there exist at most P µj − k balls B ′ = B ( y, λ − j ) ∈ C ′ LISA NAPLES satisfying µ ( B ∩ B ′ ) > . The proof of the following lemma about doubling measures and thefinite overlap condition can be found in the appendix. Lemma 1.2.
Let µ be a D -doubling measure. Then µ satisfies the finite overlap condition. The sufficient direction of the proof of Theorem A relies on the construction of a rectifiable curveusing beta numbers to determine how to connect net points in windows. A constructive algorithmfor such curves in Euclidean space was presented by Jones [Jon90] in his proof of the Analyst’sTraveling Salesman Theorem. The algorithm was adapted to infinite dimensional Hilbert space bySchul [Sch07] who removed the dimensional dependence by more carefully estimating the lengthof the curve in windows with large beta numbers. Badger and Schul [BS17] added flexibility tothe algorithm in the Euclidean setting by removing an assumption that subsequent generations ofnet points be nested. This flexibility is essential to applications in the setting of measures. See also[BNV19] by the author, Badger, and Vellis for an explicit construction algorithm of H¨older mapswhose images contain net points in Hilbert space. In the following theorem, we have removed thedimension dependence of constants in the algorithm presented in [BS17] by employing an ideafrom [Sch07] and [BNV19].
Theorem C.
Let H be a separable, infinite dimensional Hilbert space. Let C ∗ > , let x ∈ H , < δ ≤ / , and r > . Let { V k } ∞ k =0 be a sequence of nonempty, finite subsets of B ( x , C ∗ r ) such that (V1) distinct points v, v ′ ∈ V k are uniformly separated | v − v ′ | ≥ δ k r (V2) for all v k ∈ V k , there exists v k +1 ∈ V k +1 such that | v k +1 − v k | < C ∗ δ k r . (V3) for all v k ∈ V k there exists v k − ∈ V k − such that | v k − − v k | < C ∗ δ k r . Suppose that for all k ≥ and for all v ∈ V k , we are given a straight line ℓ k,v in H and a number α k,v ≥ such that sup x ∈ ( V k − S V k ) ∩ B ( v, C ∗ δ k − r ) dist( x, ℓ k,v ) ≤ α k,v δ k r , and ∞ X k =1 X v ∈ V k α k,v δ k r < ∞ . Then the sets V k converge in the Hausdorff metric to a compact set V ⊂ B ( x , C ∗ r ) , and thereexists a compact connected set such that Γ ⊂ B ( x , C ∗ r ) such that Γ ⊃ V and H (Γ) . C ∗ ,δ r + ∞ X k =1 X v ∈ V k α k,v δ k r . As illustrated by the example in [GKS10], studying measures which are carried by Lipschitzimages is a distinct problem from studying measures which are carried by Lipschitz graphs. Wedefine Lipschitz graphs in the following way. Let V be an m -dimensional plane in H . Let f : V → V ⊥ be a L -Lipschitz map. Then the set Γ = { ( v, f ( v )) : v ∈ V ) } is an L -Lipschitz graph in H .Note that the map F : V → H defined by F ( V ) = { ( v, f ( v )) : v ∈ V } is bi-Lipschitz. Lipschitz ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 7 graphs are characterized by having cone points everywhere in the following sense. Define the goodcone at x with respect to V and α by C G ( x, V, α ) := { y ∈ H : dist( y − x, V ) ≤ α | x − y |} , and the bad cone at x with respect to V and α by C B ( x, V, α ) := H \ C G ( x, V, α ) . For an L -Lipschitz graph and for x ∈ Γ , Γ ∩ C B ( x, V, α ) = ∅ where α ≥ sin(tan − ( L )) .In [MM88] Mart´ın and Mattila study sets E ⊂ R n with < H s ( E ) < ∞ and < s Let µ be a pointwise doubling measure on a separable, finite or infinite dimensionalHilbert space H . For µ -a.e. x ∈ H there is an m -plane V and an α ∈ (0 , such that (7) lim r ↓ µ ( C B ( x, r, V, α )) µ ( B ( x, r )) = 0 if and only if µ is carried by Lipschitz graphs. To explicitly see the connection to the condition (6) and condition (7), we remark that given aset E ⊂ with < H s ( E ) < ∞ , lim sup r ↓ H s ( E ∩ B ( x, r )) r s < c < ∞ for µ -a.e. x ∈ R n It follows that if lim r ↓ H s ( E ∩ C B ( x, V, α )) H s ( E ∩ B ( x, r ))) = 0 then lim r ↓ H s ( E ∩ C B ( x, V, α )) r s = 0 . For additional results on densities of measures with respect to cones, see [CKRS10], [KS08], and[KS11]. Graph rectifiability also plays a role in the study of harmonic measure. See e.g. [AAM19].1.3. Outline. The proofs of the necessary direction and the sufficient direction of Theorem A aregiven sections 2 and 3 respectively. In Section 4 we combine the results from sections 2 and 3 togive a proof of Theorem B. In Section 5, we present an example of a pointwise doubling measurewith infinite dimensional support that is carried by Lipschitz images but singular to bi-Lipschitzgraphs. In Section 6 we prove Theorem C, and finally, in Section 7, we prove Theorem D.1.4. Acknowledgment. I would like to thank my advisor Matthew Badger for insight and guid-ance throughout this project. This work was partially supported by NSF grant DMS 1650546. LISA NAPLES 2. N ECESSARY CONDITION FOR RECTIFIABILITY The goal of this section is to prove the necessary direction of Theorem A. Throughout we let H denote a separable, finite or infinite dimensional Hilbert space. We begin with a theorem aboutfinite measures that satisfy the finite overlap property. Theorem 2.1. Let ν be a finite Radon measure on H whose support is contained in the support of µ . Let Γ be a rectifiable curve, and let E ⊂ Γ such that ν ( B ( x, r )) ≥ dr for all all x ∈ E and forall < r ≤ r . Additionally, suppose that { B ∈ C µ : ν ( B ∩ E ) > } satisfies the finite overlapproperty with constants P j − k = P ( µ, j − k ) for j ≥ k . Then X B ∈ S ∞ k = l C µk β ( ν, B ) diam( B ) ν ( B ) Z E χ B ( x ) dν . H (Γ) + ν ( H \ Γ) where l is the smallest integer such that − l ≤ r . Here the implied constants depend only on d and P j − k . To prove Theorem 2.1, we will use a measure-theoretic result for weighted sums. The proof ofthe following can by found in the appendix. Lemma 2.1. Suppose that E ⊃ · · · ⊃ E k ⊃ E k +1 · · · and E = T E k . Additionally suppose ν ( E ) < ∞ , ω : E → [0 , ∞ ) , ω = 0 on E , c k ≥ , and P jk =0 c k sup x ∈ E j ω ( x ) ≤ C < ∞ . Then ∞ X k =0 c k Z E k ω ( x ) dν ( x ) = ∞ X j =0 j X k =0 c k Z E j \ E j +1 ω ( x ) dν ( x ) ≤ Cµ ( E \ E ) . We now return to the proof of Theorem 2.1. Proof. Let Γ be a rectifiable curve as specified above. We partition C µ into three subsets: C µ ∅ = { B : ν ( B ∩ E ) = 0 } , C µ Γ = ( B ∈ ∞ [ k = l C µk : ν ( B ∩ E ) > and ǫβ ( ν, B ) ≤ β Γ ( λ B ) ) , C µν = ( B ∈ ∞ [ k = l C µk : ν ( B ∩ E ) > and β Γ ( λ B ) < ǫβ ( ν, B ) ) , where restrictions on λ > and ǫ > will be specified later. Now X B ∈ S ∞ k = l C µk β ( ν, B ) diam Bν ( B ) Z E χ B ( x ) dν = X B ∈ S ∞ k = l C µl β ( ν, B ) diam B ν ( E ∩ B ) ν ( B ) ≤ ǫ − X B ∈ C µ Γ β Γ ( λ B ) diam B | {z } I + X B ∈ C µν β ( ν, B ) diam B | {z } II . We estimate the sums I and II separately. To estimate I we will invoke Theorem 1.1, the TravelingSalesman Theorem for sets in Hilbert space. In order to apply the theorem, we first need to translate ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 9 from balls centered on the carrying set X to balls centers on the rectifiable curve Γ . In doing so,we aim to establish the following bound:(8) X B ∈ C µ Γ ∩ C µk β Γ ( λ B ) diam B ≤ C X U ∈ G Γ k β Γ ( U ) diam U where k ≥ l , the constant C is independent of k , and G Γ k is a multiresolution family of balls forthe rectifiable curve Γ . The dilation factor λ for balls in G Γ k will be specified below. To showthat (8) holds, fix B ∈ C µk and let x B denote the center point. Let g ∈ B ∩ Γ which exists since ν ( B ∩ Γ) ≥ ν ( B ∩ E ) > , and let n B ∈ N k such that dist( g, n B ) ≤ − k . Now for y ∈ λ B , dist( y, n B ) ≤ dist( y, x B ) + dist( x B , g ) + dist( g, n B ) ≤ λ ( λ − k ) + λ − k + 2 − k < λ λ − k . Thus, by requiring λ ≥ λ λ , we have λ B ⊂ U = B ( n B , λ − k ) . As a consequence we cancontrol β Γ ( λ B ) with β Γ ( U ) . In particular, there is a line ℓ U such that β Γ ( U ) ≥ 12 sup y ∈ U ∩ Γ (cid:18) dist( y, ℓ U )diam( U ) (cid:19) ≥ 12 sup y ∈ λ B ∩ Γ (cid:18) dist( y, ℓ U )diam( λ B ) (cid:19) (cid:18) diam( λ B )diam( U ) (cid:19) ≥ λ λ λ β Γ ( λ B ) . Now fix a ball U ∈ G Γ k . We claim that there are at most P (cid:16) µ, ⌈ log λ λ ⌉ (cid:17) balls B ′ ∈ C µk contained in U . To see that this is the case, note that if no balls are contained in U then the boundholds trivially. Otherwise, fix B ⊂ U . It follows from triangle inequality that U ⊂ ⌈ log(2 λ /λ ) ⌉ B .By the finite overlap property there are at most P (cid:16) µ, ⌈ log λ λ ⌉ (cid:17) balls B ′ ∈ C µk such that B ′ ⊂ ⌈ log(2 λ /λ ) ⌉ B , and so there are at most P (cid:16) µ, ⌈ log λ λ ⌉ (cid:17) balls B ′ ⊂ U . This establishes inequality(8) for each k ≥ l . Now summing over all generations k and applying Theorem 1.1 (1) we concludethat X B ∈ C µ Γ β Γ ( λ B ) diam( B ) ≤ P (cid:16) µ, ⌈ log λ λ ⌉ (cid:17) λ λ λ X U ∈ G Γ β Γ ( U ) diam( U ) . H (Γ) where the symbol . depends on λ , λ , and P (cid:16) µ, ⌈ log λ λ ⌉ (cid:17) . This completes the estimate of sum I . We now begin the estimation of II . For B ∈ C µν ∩ C µk , we fix a line ℓ = ℓ B ∈ H satisfying sup z ∈ Γ ∩ λ B dist( z, ℓ ) ≤ β Γ ( λ B ) diam( λ B ) < ǫβ ( ν, B ) diam( λ B )= 2 λ ǫβ ( ν, B ) diam( B ) The first inequality follows from definition of β Γ ( λ B ) ; the second inequality follows from defini-tion of C µν . We partition B into a set of points near the line ℓ and a set of points far from the line ℓ : N ( B ) = { x ∈ B : dist( x, ℓ ) ≤ λ ǫβ ( ν, B ) diam( B ) } ,F ( B ) = { x ∈ B : dist( x, ℓ ) > λ ǫβ ( ν, B ) diam( B ) } . Using this partition of the ball B , we see that β ( ν, B ) ≤ Z N (cid:18) dist( x, ℓ )diam 2 B (cid:19) dν ( x ) ν (2 B ) + Z F (cid:18) dist( x, ℓ )diam 2 B (cid:19) dν ( x ) ν (2 B ) ≤ λ ǫ β ( ν, B ) + Z F (cid:18) dist( x, ℓ )diam(2 B ) (cid:19) dν ( x ) ν (2 B ) ≤ λ ǫ β ( ν, B ) + Z F (cid:18) dist( x, Γ ∩ λ B )diam(2 B ) (cid:19) dν ( x ) ν (2 B ) , where last inequality follows since (cid:18) dist( x, ℓ )diam 2 B (cid:19) ≤ (cid:18) dist( x, Γ ∩ λ B )diam 2 B + dist(Γ ∩ λ B, ℓ )diam 2 B (cid:19) ≤ (cid:18) dist( x, Γ ∩ λ B )diam 2 B (cid:19) + 2 (cid:18) dist(Γ ∩ λ B, ℓ )diam 2 B (cid:19) ≤ (cid:18) dist( x, Γ ∩ λ B )diam 2 B (cid:19) + 2 λ ǫ β ( ν, B ) . The choice of ℓ is used to go between the second and third lines. Now since F ⊂ B , dist( x, Γ) ≤ diam 2 B = 4 λ − k . Therefore, if we fix λ ≥ λ , then dist( x, Γ ∩ λ B ) = dist( x, Γ) . To see this explicitly, let z ∈ B and choose z Γ to be a closest point in Γ to z . Then dist( z Γ , x B ) ≤ dist( z Γ , z ) + dist( z, x B ) ≤ λ − k + λ − k < λ − k . Once λ is fixed, choose ǫ small enough to guarantee that λ ǫ < . Then we have that for fixed B β ( ν, B ) ≤ Z F (cid:18) dist( x, Γ)diam 2 B (cid:19) dµ ( x ) µ (2 B ) . To prove that II is finite, it suffices to show that X B ∈ C µν Z F (cid:18) dist( x, Γ)diam 2 B (cid:19) dµ ( x ) µ (2 B ) diam( B ) < ∞ . This sum is an improvement in that Γ is a fixed reference set which is independent of the window B .Now observe that for B ∈ C µν there exists y ∈ E ∩ B . For arbitrary z ∈ B ( y, − k ) , an applicationof the triangle inequality yields dist( z, x B ) ≤ dist( z, y ) + dist( y, x B ) ≤ λ − k , which implies that B ( y, − k ) ⊂ B . By the lower regularity assumption on points in E , µ (2 B ) ≥ µ ( B ( y, − k )) ≥ d − k = d λ diam( B ) , and, in particular, λ d ≥ diam( B ) µ (2 B ) ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 11 Using this density estimate, we conclude that X B ∈ C µν Z F (cid:18) dist( x, Γ)diam(2 B ) (cid:19) dµ ( x ) µ (2 B ) diam B ≤ λ d X B ∈ C µν Z B (cid:18) dist( x, Γ)diam(2 B ) (cid:19) dµ ( x ) ≤ λ d X B ∈ C µν B )) Z B dist ( x, Γ) dµ ( x ) ≤ λ d ∞ X k = l X B ∈ C µk k λ Z B dist ( x, Γ) dµ ( x ) ≤ d ∞ X k = l k − λ P µ Z B dist ( x, Γ) dµ ( x ) . Here the finite overlap factor P µ accounts for potential overlapping of ball in C µk . Set E k := S B ∈ C µk B , c k = 4 k − , and ω ( x ) = dist ( x, Γ) . Then j X k = l k − sup x ∈ E j ω ( x ) ≤ ∞ X k = l k − (2 − j +1 ) = j X k = l k + j − < ∞ X j = l − j − < − l . Furthermore, we verify that E k +1 ⊂ E k for each k . Let z ∈ E k +1 , and let B z = B ( x z , λ − ( k +1) ) denote a ball in C µk that contains z . By maximality of X µk , there is y z ∈ X µk such that dist( z, y z ) ≤ dist( z, x z ) + dist( x z , y z ) ≤ λ − ( k +1) + 2 − k < λ − k . In particular, this implies that z ∈ B ( y z , λ − k ) . Of course by definition of C µk , B ( y z , λ − k ) ⊂ E k , so we conclude that E k +1 ⊂ E k . Thus we may employ the following Lemma 2.1 to concludethat X B ∈ C µ Z F (cid:18) dist( x, Γ)diam 2 B (cid:19) dµ ( x ) µ (2 B ) diam(2 B ) ≤ C ( d, λ , P )) ν ( H \ Γ) . Note, in particular that C is independent of Γ . Combining our estimates of I and II , we get theestimate X B ∈ S ∞ k = l C µk β ( ν, B ) diam( B ) ν ( B ) Z E χ B ( x ) dν . H (Γ) + ν ( H \ Γ) . (cid:3) . In particular, since Γ is a rectifiable curve and ν is a finite measure, we conclude that the sum isfinite. Corollary 2.1. Let µ be a finite, lower Ahlfors d -regular Borel measure on H . Suppose that µ is D -doubling. Then Z Γ ˆ J ( µ, x ) dµ ( x ) < H (Γ) + µ ( H \ Γ) < ∞ . Proof. Recall by Theorem 1.2 that µ satisfied the finite overlap property. Then this result followsimmediately from Theorem 2.1 by setting ν = µ , observing that Z Γ ˆ J ( µ, − l , x ) dµ ( x ) = X B ∈ S ∞ k = l β ( µ, B ) diam( B ) µ ( B ) Z Γ χ B ( x ) dµ ( x ) , and applying Lemma 1.1. (cid:3) Definition 2.1 (Hausdorff density) . Let B ( x, r ) ⊂ H denote the closed ball with center x ∈ H and radius r > . We define the lower (Hausdorff) m -density at x by D m ( µ, x ) := lim inf r ↓ µ ( B ( x, r )) r m . We will show that points of zero lower density do not see rectifiable curves. This will allow usto focus on points with positive lower density for the proof of the necessary condition of TheoremA. Theorem 2.2. If µ is a pointwise doubling measure on H then µ { x ∈ H : D ( µ, x ) = 0 } ispurely unrectifiable. We will use the following two lemma. Here P denotes the -dimensional packing measure; see[Mat95, Section 5.10] for a definition. For completeness, the proofs of these lemmas are includedin the appendix. Lemma 2.2. Let E ⊂ [0 , . If f : E → H is Lipschitz then P ( f ( E )) ≤ ( Lip f ) P s ( E ) and P ( f ( E )) ≤ ( Lip f ) P ( E ) . Lemma 2.3. Let A ⊂ H be a bounded set, and suppose that there exists r > and M < ∞ suchthat for every x ∈ A and < r ≤ r µ ( B ( x, r )) ≤ M µ ( B ( x, r )) and D ( µ, x )) ≤ λ. Then µ ( A ) ≤ λ P ( A ) . We are now ready to complete the proof of Theorem 2.2. The outline follows similarly to thatof [BS15, Theorem 2.7]. However, we include details to make explicit the use of the pointwisedoubling property in the Hilbert space setting. Proof of Theorem 2.2. Suppose µ is as in the statement of the theorem, and suppose additionallythat µ is rectifiable. Set A = { x ∈ X : D ( µ, x ) = 0 } . We will show that A intersects the imageof every Lipschitz map on a set of measure zero, and hence A itself has zero measure. By Lemma2.2, for any E ⊂ [0 , , P ( f ( E )) ≤ ( Lip f ) P ( E ) < ∞ . Now let ( A ∩ f ( E )) Dj = { x ∈ A ∩ f ( E ) : µ ( B ( x, r )) ≤ Dµ ( B ( x, r )) for all < r ≤ /j } . Since µ is pointwise doubling, ∞ [ D =1 ∞ [ j =1 ( A ∩ f ( E )) Dj = A ∩ f ( E ) . Fix some D and j . Since E is bounded and f is continuous, ( A ∩ f ( E )) Dj is bounded. Now fix some λ > , and recall that D ( µ, x ) = 0 ≤ λ for all x ∈ A and, in particular, for all x ∈ ( A ∩ f ( E )) Mj .By Lemma 2.3, we have that µ (cid:0) ( A ∩ f ( E )) Dj ) (cid:1) ≤ λ P (cid:0) ( A ∩ f ( E )) Dj (cid:1) < ∞ . ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 13 Thus, letting λ → , µ (cid:0) ( A ∩ f ( E )) Dj (cid:1) = 0 for every E ⊂ [0 , and every Lipschitz function f . Hence µ (( A ∩ f ( E ))) = 0 for every E ⊂ [0 , and every Lipschitz function f . Since µ is rectifiable, we conclude that µ ( A ) = 0 . If follows immediately that for a rectifiable measure µ , D ( µ, x ) > for µ -a.e. x ∈ H , and conversely that µ { x : D ( µ, x ) = 0 } is purelyunrectifiable. (cid:3) With Theorem 2.2 established, it remains to prove the following theorem in order to obtain thenecessary condition of Theorem A. Theorem 2.3. Let µ be a pointwise doubling measure on a separable, infinite dimensional Hilbertspace H . If µ is rectifiable then ˆ J ( µ, x ) < ∞ for µ -a.e. x ∈ H. Proof. Let µ a rectifiable pointwise doubling measure on H . Since µ is rectifiable, choose a count-able family { Γ i } ∞ i =1 of rectifiable curves to which µ gives full mass, i.e., µ ( H \ S ∞ i =1 Γ i ) = 0 . As aconsequence of Theorem 2.2, µ has positive lower density µ -a.e.. This, together with the pointwisedoubling property, implies that µ gives full mass to S ∞ D =1 S ∞ m =1 S ∞ n =1 E Dm,n where E Dm,n = (cid:8) x ∈ H : µ ( B ( x, r )) ≥ − m r and µ ( B ( x, r )) ≤ Dµ ( B ( x, r )) for all r ∈ (0 , − n ] (cid:9) . Therefore, to establish the necessary direction of Theorem A, it suffices to show that ˆ J ( µ, x ) < ∞ at µ -a.e. x ∈ Γ i ∩ E Dm,n for every i , m , n , and D . To this end, fix i , m , n , and D . Set Γ = Γ i andthen set E = Γ ∩ E Dm,n . Define C µE := { B ∈ C µ : µ ( E ∩ B ) > and radius ( B ) ≤ λ − ( n +3) } , We’ll show that C µE satisfies the finite overlap property. Let B ∈ C µE ∩ C µk for some k ≥ n + 3 .Let { B i } ci =1 be the collection of balls in C µE ∩ C µj , j ≥ k , that intersect B . Then µ (4 B ) ≥ µ c [ i =1 B i ! ≥ µ c [ i =1 B ( e i , λ − ( j +1) ) ! ≥ D − N j − k c X i =1 µ (cid:0) B ( e i , N j − k λ − ( j +1) ) (cid:1) ≥ D − N j − k cµ (4 B ) . Here N j − k is the maximum number of times the ball B ( e i , − ( j +1) ) , a ball centered at a point in E and contained in B i , must be doubled to guarantee that the dilated ball contains µ (4 B ) . Note that N k − j is dependent only on the difference between j and k . We conclude that c ≤ D N j − k , so wemay take P j − k = D N j − k . Now define the measure ν by ν := µ [ C µE B. Of course Γ has finite length, and we have E ⊂ B ( x, length (Γ)) for any x ∈ E . It follows that ν has bounded support and hence, by our definition of pointwise doubling measures, ν is finite. Thus Z E ˆ J ( µ, − ( n +3) , x ) dµ ( x ) = X B ∈ C µ β ( µ, B ) diam Bµ ( B ) Z E χ B ( x ) dµ ( x )= X B ∈ C µE β ( ν, B ) diam Bν ( B ) Z E χ B ( x ) dν ( x ) . H (Γ) + ν ( H \ Γ) < ∞ where the last inequality follows from Theorem 2.1. In particular, we conclude that ˆ J ( µ, λ − ( k +3) , x ) < ∞ at µ -a.e. x ∈ E . It follows from Lemma that 1.1 ˆ J ( µ, x ) < ∞ for µ -a.e. x ∈ E. Letting i , m , n , and D vary over all natural numbers proves the result. (cid:3) 3. S UFFICIENT CONDITION FOR RECTIFIABILITY In this section we prove the sufficient condition of Theorem A. As mentioned in the introduction,the main machinery for this proof is Theorem C which is proved in sections 6.1-6.3. In orderestablish a setting in which we can apply Theorem C, we begin this section by defining a treestructure on C µ .We define the tree structure on the collection C µ to model the natural nesting structure of dyadiccubes in Euclidean space. The tree structure here is more complex than the structure for the dyadiccubes, where we can track the lineage of cubes from an initial generation, say cubes of side-length . This is because, for a fixed generation of net points X µk , we cannot in general choose a dilationof balls centered at the net points such that the balls are simultaneously pairwise disjoint and alsocovering H . To define the family structure, we rely on the following lemma. Lemma 3.1 ([Sch07], Lemma 3.19) . Given c ≤ λ and J ≥ , there exist J families of connectedsets in H such that (denoting a single family by { Q jk } k = ∞ ,j = j n k = k ,j =0 ) : (i) For every x ∈ X µk there is exists a unique j such that for B jk ∈ C µ , cB jk ⊂ Q jk for somefamily where radius ( B jk ) = λ − k . (ii) cλ − k ≤ diam Q jk ≤ · − J +1 ) cλ − k (iii) If j = j ′ then Q jk ∩ Q j ′ k = ∅ as long as Q jk and Q j ′ k belong to the same family. In this case, dist( Q jk , Q j ′ k ) ≥ − k − , (iv) If Q jk ∩ Q j ′ l = ∅ for l > k and Q jk and Q j ′ l belong to the same family then Q j ′ l ⊂ Q jk . We call the set Q jk satisfying properties (i)-(iv) the core of the ball B jk ∈ C µ . The core Q jk can bedefined in the following way. Fix k , and choose B jk ∈ C µk . If j = j ′ then dist( cB jk , cB j ′ k ) > − ( k +1) by choice of the net X µk . Set Q jk, := cB jk ,Q jk,i +1 := Q jk,i ∪ [ cB j ′ k +( i +1) J ∩ Q jk,i = ∅ cB j ′ k +( i +1) J ,Q jk := lim i →∞ Q jk,i . Then we define the k th family of cores to be Q k := (cid:8) Q jk + iJ : i ∈ N (cid:9) , and we denote the collection of all cores by Q := ∞ [ k = k Q k . ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 15 We remark that the construction of these cores depends on the choices of the constant c and J . Forballs that belong to the same family there are intrinsic tree structures given by inclusion. We usethe tree structures on Q k to define a tree structure on the balls in C µ . In particular, • for l = k + J , if Q jk ∩ Q j ′ l = ∅ then Q j ′ l ⊂ Q jk , and we say that B j ′ l is a child of B jk ( B jk ≻ B j ′ l ); • for k = l − J , if there exists j such that Q jk ∩ Q j ′ l = ∅ then Q jk ⊃ Q j ′ l , and we say that B jk is the parent of B j ′ l , ( B j ′ l ) ↑ = B jk ; • for l ≥ k + iJ and i ≥ , if there is Q jk such that Q jk ∩ Q j ′ l = ∅ and Q jk and Q j ′ l belong tothe same family then we say that B j ′ l is a descendant of B jk and B jk is an ancestor of B j ′ l .We extend the parent, child, and descendant relationships to net points x jk and x j ′ l in the obviousway. By property (iii), when a ball or net point has a parent, the parent is unique. We say acollection T ⊂ C µ is a tree if(1) there exists a unique B ∈ T such that for every ball B ∈ T , B is a descendant B . Wedenote the ball B by Top ( T ) and we call B the top of tree T ;(2) for every B ∈ T \ { B } , B ↑ ∈ T .A branch of T is a sequence of balls B ≻ B ≻ B ≻ · · · such that each B i ∈ T . A branch is finite if there is some B t ∈ T such that for all B i ∈ T \ { B t } , B t B i . That is, nochild of B t is contained in the tree. If a branch is not finite then it is infinite. We define the leaves of the tree T to be the setLeaves ( T ) := [ n lim i →∞ B i : B ≻ B ≻ B ≻ ... is an infinite branch of T o . Here the limit is taken to be the intersection of nested sets, lim i →∞ ∩ ∞ j =0 B j . Now we specify λ > (1 − − J ) − . This specification allows us to prove the following containment of childreninside of parent balls. Lemma 3.2. If B j ′ l ≺ B jk , then B j ′ l ⊂ B jk .Proof. Since B j ′ l ≺ B jk , (4 λ ) − B j ′ l ∩ Q jk, = ∅ . This implies that (cid:12)(cid:12)(cid:12) x B j ′ l − x B jk (cid:12)(cid:12)(cid:12) ≤ l X i =1 − k − iJ ≤ − k (cid:18) − − J (cid:19) . Fix y ∈ B j ′ l , and observe that (cid:12)(cid:12)(cid:12) y − x B jk (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) y − x B j ′ l (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) x B j ′ l , x B jk (cid:12)(cid:12)(cid:12) ≤ (cid:0) λ − J + (1 − − J (cid:1) − )2 − k . By the choice of λ we have we have λ (1 − − J ) > − − J , so ( λ − J + (1 − − J ) − ) < λ − k .It follows that y ∈ B jk . (cid:3) Now that a tree structure has been defined on C µ , we will show that a rectifiable curve can bedrawn through the leaves of a tree. We begin with a lemma that relates the center of mass of a setto its L beta number. This is an adaptation of [Ler03, Lemma 6.4]. Lemma 3.3. Let µ be a Radon measure on H , let E be a Borel set of positive diameter such that < µ ( E ) < ∞ , and let z E := Z E z dµ ( x ) µ ( E ) ∈ H denote the center of mass of E with respect to µ . For every straight line ℓ in H , dist( z E , ℓ ) ≤ β ( µ, E, ℓ ) diam E. Proof. For every affine subspace ℓ in H , the function dist( · , ℓ ) is convex. Thus, dist( z E , ℓ ) = dist (cid:18)Z E z dµ ( z ) µ ( E ) , ℓ (cid:19) ≤ Z E dist( z, ℓ ) dµ ( z ) µ ( E ) = β ( µ, E, ℓ ) (diam E ) by Jensen’s inequality. (cid:3) We will also need the following lemma which says that a compact connected set of finite lengthis a Lipschitz image. Lemma 3.4 ([Sch07], Lemma 3.7) . If Γ ⊂ H is a closed, connected set such that H (Γ) < ∞ ,then there exists a Lipschitz map f : [0 , → H such that Γ = f ([0 , . Moreover, f can be foundsuch that | f ( s ) − f ( t ) | ≤ H (Γ) | s − t | for all ≤ s, t ≤ . Lemma 3.5 (Drawing rectifiable curves through the leaves of uniformly doubling trees, cf. [BS17]Lemma 7.3) . Let µ be a finite measure on H and let ≤ d T < ∞ . If T is a tree of balls from themultiresolution family C µ such that (9) µ ( B ↑ ) ≤ d T µ ( B ) for all B ∈ T and S ( µ, T ) := X B ∈T β ( µ, B ) diam B < ∞ . Additionally, suppose that T satisfies the finite overlap property. Then there exists a rectifiablecurve Γ in H such that Γ ⊃ Leaves ( T ) and H (Γ) . diam Top ( T ) + d J T S ( µ, T ) . Proof. By dilating and translating as needed, we may assume that Top ( T ) = B (0 , λ ) =: B .Deleting irrelevant balls from T , we may also assume that every ball B ∈ T belongs to an infinitebranch of T . Our goal is apply Theorem C. Set parameters C ∗ = 5 · J , r = diam( Top ( T )) = 2 λ ,and δ = 2 − J where J is as in Lemma 3.1. For each B ∈ T , let z B = R B z dµ ( z ) µ ( B ) denote the centerof mass of B , and for k ≥ , set Z k = { z B : B ∈ T ∩ C µk } . Choose V k to be any maximal δ k r -separated subset of Z k , and fix x ∈ B . Then V k ⊂ Z k ⊂ Top ( T ) ⊂ B ( x , r ) ⊂ B ( x , C ∗ r ) Clearly, V k satisfies (V1) of Theorem C. It remains to verify that (V2) and (V3) hold. We beginwith (V2). Let k ≥ and let v ∈ V k , say v = z B for some B ∈ T ∩ C µk . As a consequence of ourassumption that every ball in T belongs to an infinite branch of T , there exists R ∈ T such that R ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 17 is a child of B . By maximality of V k +1 , there is v ′ = z P ∈ V k +1 for some P ∈ T ∩ C µk +1 such that | z R − z P | < δ k +1 r . It follows that | z B − z P | ≤ | z B − z R | + | z R − z P |≤ | z B − x B | + | x B − x R | + | x R − z R | + | z R − z P |≤ 12 diam( B ) + 12 diam( Q B ) + 12 diam( Q R ) + 12 diam( R ) + δ k +1 ≤ · δ k r < C ∗ δ k r . Thus condition (V2) is satisfied.Finally, to check condition (V3), let k ≥ and let v ∈ V k , say v = z B for some B ∈ T ∩ C µk .Let R denote the parent of B which necessarily belongs to T . By maximality, there exists v ′ = z P ∈ V k − for some P in the same generation as R with | z P − z R | < δ k − r . It follows that | z B − z P | ≤ | z B − z R | + | z R − z P |≤ | z B − x B | + | x B − x R | + | x R − z R | + | z R − z P |≤ 12 diam( B ) + 12 diam( Q B ) + 12 diam( Q R ) + 12 diam( R ) + δ k − ≤ · J r δ k = C ∗ r δ k . So (V3) is satisfied as well.Fix k ≥ and v ∈ V k , and choose a ball B k,v ∈ T such that v = z B k,v . That is, B k,v is the ballin C µk whose center of mass is v . For each k ≥ and v ≥ V k , let ˆ B k,v ∈ T denote the minimalancestor of B k,v satisfying • ˆ B k,v ⊃ B k,v ; • for every v ′ ∈ V k ∩ B ( v, C ∗ δ k − r ) and j ∈ { k − , k } , ˆ B k,v ⊃ B j,v ′ Now C ∗ δ k − r = 66 · J C ∗ δ k r < J λ δ k , so diam ˆ B k,v diam B k,v ≤ J for all j ∈ { k − , k } and v ′ ∈ V k ∩ B ( v, C ∗ δ k − r ) . Furthermore, from assumption (9) we getthat,(10) µ (2 ˆ B k,v ) µ (2 B j,v ′ ) ≤ d J T for all j ∈ { k − , k } and v ′ ∈ V j ∩ B ( v, C ∗ δ k − r ) . Next, let k ≥ and let v ∈ V k . Choose ℓ k,v to be any straight line in X such that(11) β ( µ, B k,v , ℓ k,v ) ≤ β ( µ, B k,v ) . Such line exists by definition of β -number. By combining estimate (10) and (11) we see that β ( µ, B j,v ′ , ℓ j,v ′ ) diam B j,v ′ = Z B j,v ′ (cid:18) dist( x, ℓ k,v )diam(2 B j,v ′ ) (cid:19) dµ ( x ) µ (2 B j,v ′ ) ! diam( B j,v ′ ) ≤ Z B k,v dist( x, ℓ k,v )diam ˆ B k,v ! dµ ( x ) µ (2 B j,v ′ ) diam B k,v (cid:18) diam B j,v ′ diam B k,v (cid:19) diam ˆ B k,v diam B j,v ′ ! µ (2 ˆ B k,v ) µ (2 B j,v ′ ) ! ≤ J d J T β ( µ, B k,v , ℓ k,v ) diam B k,v ≤ (4 d T ) J β ( µ, B k,v ) diam B k,v =: α k,v δ k r for all j ∈ { k, k − } and all v ′ ∈ B ( v, C ∗ δ k − r ) . This verifies the remaining hypothesis ofTheorem C. Now since T satisfies the finite overlap property, the number of times a ball B ∈ T appears as ˆ B k,v is bounded, and the bound depends on at most the finite overlap constant P ( T , J ) . We conclude that there exists a compact, connected set Γ ⊂ H such that H (Γ) . C ∗ r + ∞ X k =1 X v ∈ V k α k,v δ k r . diam Top ( T ) + d J T S p ( µ, T ) , and Γ ⊃ V = lim k →∞ V k . By Lemma 3.4, Γ is a rectifiable curve. It remains to check that Γ ⊃ Leaves ( T ) .Let y ∈ Leaves ( T ) , say y = lim k →∞ y k for a sequence of points y k ∈ B k corresponding toan some infinite branch B ≻ B ≻ B ≻ ... of T . Let z k = z B k denote the center of mass of B k and let v k ∈ V k be any point which minimizes distance to z k . By maximality of the net V k , | z k − v k | ≤ δ k r . Furthermore, since both z k and y k live in B k , | z k − y k | ≤ diam B k = δ k r .Combining these estimates, we get | v k − y | ≤ | v k − z k | + | z k − y k | + | y k − y | ≤ · δ k r + | y k − y | for all k ≥ . Thus y = lim k →∞ v k ∈ lim k →∞ V k ⊂ Γ . Since y ∈ Leaves ( T ) arbitrary, we conclude that Γ ⊃ Leaves ( T ) . (cid:3) We now prove a lemma which is an adaptation of [BS17, Lemma 5.6] to the setting of trees onwhich µ satisfies a doubling property. Let T be a tree of balls in C µ , and define a µ -normalizedsum function by ˆ S T ,b ( µ, x ) := X B ∈T b ( B ) µ ( B ) χ B ( x ) for all x ∈ H. We interpret / and / ∞ . The following result hold. Lemma 3.6 (Localization lemma for doubling tree) . Let T ⊂ C µ be a tree, and suppose thatthere exists a constant D T such that µ ( B ) ≤ D T µ ( aB ) for every B ∈ T where a is some fixedconstant satisfying some < a ≤ c . Here c is as in Lemma 3.1. Let b : T → [0 , ∞ ) . Then for all N < ∞ , and ǫ > , there exists a partition of T into a set Good ( T , N, ǫ ) of good balls and a setBad ( T , N, ǫ ) of bad balls with the following properties. (i) Either Good ( T , N, ǫ ) = ∅ or Good ( T , N, ǫ ) is tree of balls from C µ withTop ( Good ( T , N, ǫ )) = Top ( T ) . ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 19 (ii) Every child of a bad ball is a bad ball: if B and R belong to T , R ∈ Bad ( T , N, ǫ ) and B ≺ R , then B ∈ Bad ( T , N, ǫ ) . (iii) The set E := { x ∈ Top ( T ) : S T ,b ( µ, x ) ≤ N } and E ′ := E ∩ Leaves ( Good ( T , N, ǫ )) have comparable measures: µ ( E ′ ) ≥ (1 − ǫµ ( Top ( T ))) µ ( E ) . (iv) The sum of the function b over the good cubes is finite X B ∈ Good ( T ,N,ǫ ) b ( B ) < N D T ǫ . Proof. Suppose that T , µ , b , N , ǫ , E , and E ′ are as given in the statement of the lemma. If µ ( E ) = 0 then we may declare every ball B ∈ T to be a bad ball, and the conclusion of the lemmaholds trivially. Therefore, suppose that µ ( E ) > . Declare a ball B ∈ T to be a bad ball if thereexists a ball B ′ ∈ T such that B is a descendant of B ′ and B ′ satisfies µ ( E ∩ B ′ ) ≤ ǫµ ( E ) µ ( Q B ′ ) where Q B ′ is the core of the ball B ′ . We call B a good ball if B is not a bad ball. Properties(i) and (ii) are immediately satisfied by the definitions of good and bad balls. To check property(iii) we remark that E \ Leaves ( Good ( T , N, ǫ )) ⊂ E ∩ S B ∈ Bad ( T ,N,ǫ ) B. Let Bad M ( T , N, ǫ ) ⊂ Bad ( T , N, ǫ ) denote the set of maximal bad balls. That is, B ∈ Bad M ( T , N, ǫ ) if no ancestor of B is a bad ball. Then µ ( E \ E ′ ) ≤ µ E ∩ [ B ∈ Bad ( T ,N,ǫ ) B ≤ µ E ∩ [ B ∈ Bad M ( T ,N,ǫ ) B ≤ X B ∈ Bad M ( T ,N,ǫ ) µ ( E ∩ B ) ≤ ǫµ ( E ) X B ∈ Bad M ( T ,N,ǫ ) µ ( Q B ) ≤ ǫµ ( E ) µ [ B ∈ Bad ( T ,N,ǫ ) Q B ≤ ǫµ ( E ) µ ( Top ( T )) . Note that for the second inequality we use Lemma 3.2 and for the penultimate inequality we usethat the cores of the maximal balls are disjoint by Property (iv) of Lemma 3.1. Thus µ ( E ′ ) = µ ( E ) − µ ( E \ E ′ ) ≥ (1 − ǫµ ( Top ( T ))) µ ( E ) so property (iii) holds.Before we begin the proof of (iv),we recall that by definition of T and by the construction ofcores Q B ,(12) µ ( B ) ≤ D T µ ( aB ) ≤ D T µ ( Q B ) . Finally, since ˆ S T ,b ( µ, x ) ≤ N for all x ∈ E , N µ ( E ) ≥ Z E ˆ S T ,b ( µ, x ) dµ ( x ) ≥ Z E X B ∈T b ( B ) µ ( B ) χ B ( x ) dµ ( x ) ≥ D T X B ∈T b ( B ) µ ( E ∩ B ) µ ( Q B ) ≥ ǫD T µ ( E ) X B ∈ Good ( T ,N,ǫ ) b ( B ) . The second to last inequality follows by (12), and the last equality holds because balls in Good ( T , N, ǫ ) satisfy µ ( E ∩ B ) > ǫµ ( E ) µ ( Q B ) . We conclude that P B ∈ Good ( T ,N,ǫ ) b ( B ) ≤ N D T /ǫ . (cid:3) Theorem 3.1. Let µ be a Radon measure on H . Then the measure µ (cid:26) x ∈ H : lim sup r ↓ µ ( B ( x, r )) µ ( B ( x, r )) < ∞ and ˆ J ( µ, x ) < ∞ (cid:27) is -rectifiable.Proof. Fix x ∈ H such that lim sup x ↓ µ ( B ( x, r )) /µ ( B ( x, r )) < ∞ . There exists ≤ ω x < ∞ and r x > such that µ ( B ( x, r )) ≤ ω x µ ( B ( x, r )) for all < r < r x . Let a ′ be an integer suchthat a ′ ≥ /c , where c is as in Lemma 3.1. Then µ ( B ) ≤ µ ( B ( x, diam( B )) ≤ ( a ′ +1) ω x µ (cid:16) B (cid:16) x, radius (cid:16) − ( a ′ +1) B (cid:17)(cid:17)(cid:17) ≤ D λ ,ω x µ ( cB ) for every B ∈ C µ such that x ∈ − ( a ′ +1) B and radius ( B ) < r x . A similar series of inequalitiesshows that if B ′ ≺ B then B ⊂ J +2 B ′ and µ ( B ) ≤ µ (2 J +2 B ′ ) ≤ d ω x µ ( B ′ ) for some constant d ω x depending on ω x . Thus, x belongs to the leaves of the tree T x = (cid:8) B ∈ C µ : B ≺ B x , µ ( R ) ≤ D λ ,ω x µ ( cR ) , µ ( R ↑ ) ≤ d ω x µ ( R ) for all R ∈ C µ s.t. B ≺ R ≺ B x (cid:9) , where B x ∋ x is defined to be a maximal ball in a family satisfying radius (8 B ) < r x . By Lemma3.2, x ∈ Top ( T x ) ∩ Leaves ( T x ) . Now since each tree T x is determine by B x ∈ C µ and C µ iscountable, we can enumerate the trees (cid:26) T x : lim sup x ↓ µ ( B ( x, r )) /µ ( B ( x, r )) < ∞ (cid:27) = {T x i , i = 1 , , , ... } for x i ∈ spt µ . Thus, (cid:26) x ∈ H : lim sup x ↓ µ ( B ( x, r )) /µ ( B ( x, r )) < ∞ and ˆ J ( µ, x ) < ∞ (cid:27) ⊂ ∞ [ i =1 ∞ [ j =1 { x ∈ Top ( T x i ) : ˆ J ( µ, x ) ≤ j } , ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 21 so it suffice to prove that the measure µ A y,N is -rectifiable for arbitrary y in the carry set X such that ˆ J ( µ, y ) ≤ N where A y,N := { x ∈ Top ( T y ) : ˆ J ( µ, x ) ≤ N } . Fix such y and N . Set η y := µ ( Top ( T y )) . Given < ǫ < η y , let T y,N,ǫ := Good ( T y , N, ǫ ) ⊂ T y denote the tree given by Lemma 3.6 with T = T y , b ( Q ) = β ( µ, B ) diam B , and a = c . Thenby Lemma 3.6, S ( µ, T y,N,ǫ ) < N D T y,N,ǫ /ǫ ) and µ ( A y,N ∩ Leaves ( T y,N,ǫ )) ≥ (1 − ǫη y ) µ ( A y,N ) . By Lemma 3.5, there exists a rectifiable curve Γ y,N,ǫ in H such that Γ y,N,ǫ captures a significantportion of the mass of A y,N : µ ( A y,N \ Γ y,N,ǫ ) ≤ µ ( A y,N ) − µ ( A y,N ∩ Γ y,N,ǫ ) ≤ µ ( A y,N ) − (1 − ǫη y ) µ ( A y,N ) = ǫη y µ ( A y,N ) . Finally, for k ≥ , choose < ǫ k < η y such that lim k →∞ ǫ k = 0 . Then µ A y,N \ ∞ [ k =1 Γ y,N,ǫ k ! ≤ inf k ≥ µ ( A y,N \ Γ y,N,ǫ ) ≤ η y µ ( A y,N ) inf k ≥ ǫ k = 0 . We conclude that µ A y,N is -rectifiable. This completes the proof. (cid:3) An immediate corollary of this result is the sufficient direction of Theorem A.4. P ROOF OF T HEOREM BWe are now ready to prove the decomposition result, Theorem B. Proof of Theorem B. Let µ be a pointwise doubling measure on an infinite dimensional Hilbertspace H , and partition H into two sets: R = { x ∈ H : ˆ J ( µ, x ) < ∞} and P = { x ∈ H : ˆ J ( µ, x ) = ∞} . It is clear that both R and P are Borel sets. Since R and P partition H , we have µ = µ R + µ P and µ R ⊥ µ P. The decomposition µ = µ rect + µ pu is unique (see [BS17, Theorem 1.2]), so to prove Theorem B itsuffices to show that µ R is rectifiable and µ P is purely unrectifiable. By Theorem 3.1, µ R is -rectifiable. Additionally, µ P ≤ µ { x ∈ H : D ( µ, x ) = 0 } + µ { x ∈ H : D ( µ, x ) > and ˆ J ( µ, x ) = ∞} . By Theorem 2.2 µ { x ∈ H : D ( µ, x ) = 0 } is purely unrectifiable, and by Theorem 2.3 µ { x ∈ H : D ( µ, x ) > and ˆ J ( µ, x ) = ∞} is purely unrectifiable. Therefore, µ P is also purelyunrectifiable. This completes the proof of Theorem B. (cid:3) 5. A N EXAMPLE OF POINTWISE DOUBLING MEASURE WITH INFINITE DIMENSIONALSUPPORT In this section we construct a pointwise doubling measure µ which has infinite dimensionalsupport, is carried by Lipschitz images, and assigns zero measure to every bi-Lipschitz image.To construct the measure, we build off a construction by Garnett, Killip, and Schul of a doublingmeasure on R n which is carried by Lipschitz images but singular to bi-Lipschitz images. Theorem 5.1 (Garnett, Killip Schul [GKS10]) . For n ≥ there exists a -rectifiable doublingmeasure ν n with spt ν n = R n . Let ν = ν be as in [GKS10], and let C ν denote the doubling constant. Let V ∈ H be a twodimensional linear plane. Fix a basis on H so that for x ∈ V , x = ( a , a , , , . . . ) for some a , a ∈ R . By the separability of H choose a dense collection { x i } ∞ i =1 of V ⊥ , the orthogonalcomplement of V . Set V i = V + x i . We identify each V i , i = 0 , , , , ... with R using the map π i : V i → R defined by π i (( a , a , a , ... )) = ( a , a ) . Let { c i } ∞ i =0 be a summable sequence of positive numbers, i.e., c i > for each i and P ∞ i =1 c i < ∞ . Then set µ := P ∞ i =0 c i ν i where ν i ( E ) := ν ( π i ( E ∩ V i )) . In particular, for y ∈ V i ν j ( B ( y, r )) := ν ( B ( π j ( y ) , S ijr )) , where S ijr := (q r − dist ( V i , V j ) , if dist( V i , V j ) < r otherwise.Since ν is rectifiable, µ is also rectifiable. That µ is finite on bounded sets follows from the summa-bility of the sequence { c i } together with the fact that ν is finite on bounded sets. Furthermore, µ -a.e. y is an element of some point V i . Now fix some such y and denote by V i y the plane that containsthis y . Choose N y > i y such that P ∞ N y +1 c i ≤ c iy . For r < min { ,...,N y }\{ i y } dist( V i y , V i ) > , µ ( B ( y, r )) = N y X i =1 c i ν i ( B ( y, r )) + ∞ X i = N y +1 c i ν i ( B ( y, r ))= c i y ν i y ( B ( y, r )) + ∞ X i = N y +1 c i ν ( B ( π i ( y ) , S i y i r )) ≤ c i y ν i y ( B ( y, r )) + ν i y ( B ( y, r )) ∞ X i = N y +1 c i ≤ c i y ν i y ( B ( y, r )) + c i y ν i y ( B ( y, r )) ≤ c i y ν i y ( B ( y, r )) + µ ( B ( y, r ))2 . It follows that µ ( B ( y, r )) ≤ c i y ν i y ( B ( y, r )) = 2 c i y ν ( B ( π ( y ) , r )) ≤ c i y C ν ν ( B ( π ( y ) , r )) ≤ C ν µ ( B ( y, r )) . Thus for every µ -a.e. y , lim sup r ↓ µ ( B ( y, r )) µ ( B ( y, r )) ≤ C ν ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 23 so µ is a pointwise doubling measure. By the density of the collection { x i } ∞ i =1 , and since thecoefficients c i were chosen to be nonzero, spt ( µ ) = H .6. D RAWING CURVES THROUGH NETS : AN A NALYST ’ S T RAVELING S ALESMAN A LGORITHM In this section we prove Theorem C. The proof follows the same outline as the proof of Propo-sition 3.6 in [BS17]. We provide full details to the portions of the proof that require adaptationsto the setting of infinite dimensional Hilbert space, and we refer the reader to appropriate sectionsin [BS17] for portions that follow identically. The required adaptations, which serve to remove di-mension dependence, draw on ideas from [Sch07] and [BNV19]. We begin by restating TheoremC for convenience. Theorem C. Let H be a separable, infinite dimensional Hilbert space. Let C ∗ > , let x ∈ H , < δ ≤ / , and r > . Let { V k } ∞ k =0 be a sequence of nonempty, finite subsets of B ( x , C ∗ r ) such that (V1) distinct points v, v ′ ∈ V k are uniformly separated | v − v ′ | ≥ δ k r ; (V2) for all v k ∈ V k , there exists v k +1 ∈ V k +1 such that | v k +1 − v k | < C ∗ δ k r ; (V3) for all v k ∈ V k there exists v k − ∈ V k − such that | v k − − v k | < C ∗ δ k r . Suppose that for all k ≥ and for all v ∈ V k , we are given a straight line ℓ k,v in H and a number α k,v ≥ such that sup x ∈ ( V k − S V k ) ∩ B ( v, C ∗ δ k − r ) dist( x, ℓ k,v ) ≤ α k,v δ k r , and ∞ X k =1 X v ∈ V k α k,v δ k r < ∞ . Then the sets V k converge in the Hausdorff metric to a compact set V ⊂ B ( x , C ∗ r ) , and thereexists a compact connected set such that Γ ⊂ B ( x , C ∗ r ) such that Γ ⊃ V and H (Γ) . C ∗ ,δ r + ∞ X k =1 X v ∈ V k α k,v δ k r . As in [BS17], we prove Theorem C in three parts. In section 6.1 we construct sets Γ k byconnecting vertices in V k with straight line segments. In section 6.2 we verify that the sets Γ k are connected. Finally, in section 6.3 we justify the length estimate on the limiting set. For easeof notation, we assume that r = 1 throughout our construction of the curves. We will need thefollowing two lemmas. Lemma 6.1. Let B ⊂ H be a bounded set and let V , V , ... be a sequence of nonempty finitesubsets of B . If the sequence satisfies (V2) and (V3) for some C ∗ > and r > then V k converges in the Hausdorff metric to a closed set V ⊂ B . Lemma 6.2 ([BS17, Lemma 8.3]) . Suppose that V ⊂ R n is a -separated set with V ≥ andthere exist lines ℓ and ℓ and a number ≤ α ≤ / such that dist( v, ℓ i ) ≤ α for all v ∈ V and i = 1 , . Let π i denote the orthogonal projection onto ℓ i . There exist compatible identifications of ℓ and ℓ with R such that π ( v ′ ) ≤ π ( v ′′ ) if and only if π ( v ′ ) ≤ π ( v ′′ ) for all v ′ , v ′′ ∈ V . If v and v areconsecutive points in V relative to the ordering of π ( V ) , then H ([ u , u ]) ≤ (1 + 3 α ) · H ([ π ( u ) , π ( u )]) for all [ u , u ] ⊂ [ v , v ] . Moreover, H ([ y , y ]) ≤ (1 + 12 α ) · H ([ π ( y ) , π ( y )]) for all [ y , y ] ⊂ ℓ . Lemma 6.1 is an analogue to [BS17, Lemma 8.2] in the setting of Hilbert space. However, wepresent a different proof technique to overcome to fact the closed, bounded sets are not necessarilycompact in Hilbert space. The proof can be found in the appendix. Although H may be infinitedimensional, we will apply Lemma 6.2 to V k for each k . Since V k is a finite collection of pointswe may think of V k as being embedded in R n k where n k is at least the cardinality of V k .We fix a parameter < ǫ < / so that the conclusions of Lemma 6.2 hold for α = 2 ǫ . Thisparameter will be used throughout our definition of Γ k . For each k , we partition V k into a set avertices with α k,v less than ǫ and a set of vertices with α k,v greater than or equal to ǫ . That is, weset V k = V Flat k S V Non-flat k where V Flat k = { v ∈ V k : α k,v < ǫ } and V Non-flat k = { v ∈ V k : α k,v ≥ ǫ } .Our construction of Γ k near a vertex v will depend on whether v ∈ V Flat k or v ∈ V Non-flat k .6.1. Description of curves. We construct curves Γ k to be the union of finitely many closed setswhich take two forms.(1) edges [ v ′ , v ′′ ] : closed line segments between vertices v ′ , v ′′ ∈ V k .(2) bridges B [ j, w ′ , w ′′ ] : closed sets that connect vertices w ′ , w ′′ ∈ V j for some k ≤ j ≤ k andpass through vertices of generation j ′ nearby w ′ and w ′′ for every j ′ > j . More explicitly,for j ≥ k and v ∈ V j define an extension e [ j, v ] in the following way. Given v = v , pick asequence of vertices v , v , ..., inductively so that v is a vertex in V j +1 that is closest to v , v is a vertex in V j +2 that is closest to v , etc. Then define e [ j, v ] := S ∞ i =0 [ v i , v i +1 ] . Onceextensions have been chosen, for each generation j ′ ≥ j we define the bridge B [ j, w ′ , w ′′ ] by B [ j, w ′ , w ′′ ] := e [ j, w ′ ] ∪ [ w ′ , w ′′ ] ∪ e [ j, w ′′ ] . If an edge [ v ′ , v ′′ ] is included in Γ k , then | v ′ − v ′′ | < C ∗ δ k − . We will store edges constructed ingeneration k in a set denoted by Edge ( k ) . We will store each bridge in one of two sets: Bridge Flat ( k ) or Bridge Non-flat ( k ) . We will add bridges to Bridge Flat ( k ) when we are constructing a portion of Γ k nearby a vertex v satisfying α k,v < ǫ , and we will add bridges to Bridge Non-flat ( k ) when we areconstructing Γ k for vertices v with α k,v ≥ ǫ . We denote the set of all bridges by Bridge ( k ) := Bridge Flat ( k ) ∪ Bridge Non-flat ( k ) . Bridges are frozen in that if a bridge B [ k, v ′ , v ′′ ] appears in Γ k for some k then that B [ k, v ′ , v ′′ ] also appears in Γ k ′ for all k ′ ≥ k . We will need the followingdefinition of semi-flat vertices to build Γ k near non-flat vertices. Definition 6.1 (Semi-flat vertex) . For k ≥ k + 1 , we call a vertex y ∈ V k − a semi-flat vertex if α k − ,y ≥ ǫ and there exists a vertex v ∈ V k such that | y − v | ≤ C ∗ δ k − and α k,v ≤ ǫ . ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 25 Given a semi-flat vertex y ∈ V k − , we can choose vertex v ∈ V Flat k such that | v − y | ≤ C ∗ δ k − .Then since B ( y, C ∗ δ k − ) ⊂ B ( v, C ∗ δ k − ) , there exists a natural linear ordering on the pointsin V k − ∩ B ( y, C ∗ δ k − ) defined in terms of projection onto ℓ k,v . We define a set S k of edges em-anating from semi-flat vertices in V k in the following way. Fix a semi-flat vertex y , and enumeratethe points in V k − ∩ B ( y, C ∗ δ k − ) from left to right: y − l , ..., y − , y = y, y , ..., y m . Add edges [ y i , y i +1 ] to S k for ≤ i ≤ m − until | y − i − y i +1 | ≥ C ∗ δ k − or until y i +1 doesnot exist. We symmetrically add edges to S k between vertices to the left of y .If V k = 1 for infinitely many k then we can choose Γ k to be a singleton and the theorem holdstrivially. Thus let k ≥ be the smallest index such that V k ≥ for all k ≥ k . It suffices thento describe the construction of Γ k for k ≥ k . We will first describe the construction of Γ k . Thesubsequent constructions follow by induction on k .6.1.1. Base Case: The construction of Γ k . We claim that for any v ∈ V k , V k ⊂ B ( v, C ∗ δ k ) ⊂ B ( v, C ∗ δ k − ) . To see that this is true, recall that by definition of k , there is a unique element w ∈ V k − . Additionally, by (V3), for any v, v ′ ∈ V k , we have | v − w | ≤ C ∗ δ k and | v ′ − w | ≤ C ∗ δ k . Hence, | v − v ′ | ≤ | v − w | + | w − v ′ | ≤ C ∗ δ k . Now suppose that V Flat k = ∅ , and fix some element v in the set. By Lemma 6.2 there exists a linearordering on V k , v − l , ..., v − , v , v , ...v m according to orthogonal projection onto the line ℓ k ,v . We connect v i to v i +1 with an edge [ v i , v i +1 ] for all − l ≤ i ≤ m . We store each edge in Edge ( k ) .Suppose instead that V Flat k = ∅ . If there exists v ∈ V Non-flat k which is semi-flat with respect tosome y ∈ V k +1 then the vertices in V k can be ordered according to projection on ℓ k +1 ,y , and weadd edges as in the case when V Flat k = ∅ . Otherwise, enumerate the vertices in V k arbitrarily as v , v , ...., v m and connect v i to v i +1 with the edge [ v i , v i +1 ] for ≤ i ≤ m − . We store each edgein Edge ( k ) .In any case, we define Γ k to be the union of edges in Edge ( k ) .6.1.2. Inductive Case: The construction of Γ k from Γ k − . Suppose Γ k ,..., Γ k − have been definedfor some k ≥ k + 1 . To define the next set Γ k we describe the construction of Γ k,v , the new partof Γ k nearby v for every v ∈ V k . We will first describe the construction of Γ k,v for v ∈ V Flat k , andwe will subsequently describe the construction of Γ k,v for v ∈ V Non-flat k . We refer to constructionnear vertices in V Flat k as “ Case F construction” and construction near vertices in V Non-flat k as “ CaseN construction.” As mentioned above, edges added in each stage of construction are include inEdge ( k ) , and bridges added during Case F are included in Bridge Flat ( k ) whereas bridges addedduring Case N are included in Bridge Non-flat ( k ) . Case F Construction. This step of construction follows identically to the case of vertices v satis-fying α k,v < ǫ in Section 8.2 of [BS17] with C ∗ δ k in place of C ∗ − k and C ∗ δ k − in placeof C ∗ − k . We include further exposition in order to introduce notation that will be used later inthe paper.Fix v ∈ V Flat k . Identify ℓ k,v with R (and pick a direction “left” and “right”). Let π k,v denoteorthogonal projection onto ℓ k,v . Since α k,v ≤ ǫ , by Lemma 6.2 and (V1), both V k ∩ B ( v, C ∗ δ k − ) and V k − ∩ B ( v, C ∗ δ k − ) can be arranged linearly along ℓ k,v . Set v = v ∈ V k and let v − l , ..., v − , v , v , ..., v m denote the vertices in V k ∩ B ( v, C ∗ δ k − ) arranged from left to right relative to the order of π k,v ( v i ) in ℓ k,v . We will first describe the construction of the “right half”, Γ Rk,v , of Γ k,v . Startingwith v and working right, include each closed line segment [ v i , v i +1 ] as an edge in Γ Rk,v until oneof the following holds: • | v i +1 − v i | ≥ C ∗ δ k − • v i +1 / ∈ B ( v, C ∗ δ k − ) • v i +1 is undefined.Let t ≥ denote the number of edges that were included in Γ Rk,v . We consider three subcases: Case F-NT: If t ≥ then the vertex v is non-terminal to the right, and we are done describing Γ Rk,v . Case F-T: If t = 0 then v is a terminal vertex . We determine the construction of Γ k be studying thebehavior of Γ k − nearby v . Let w v be a vertex in V k − that is closest to v . Enumerate the verticesin V k − ∩ B ( v, C ∗ δ k − ) starting from w v and moving right w v = w v, , w v, , ..., w v,s . Let w v,r denote the rightmost vertex in V k − ∩ B ( v, C ∗ δ k − ) . There are two alternatives whichdetermine our subcases. Case F-T1: If r = s or if | w v,r − w v,r +1 | ≥ C ∗ δ k − , set Γ Rk,v = { v } . Case F-T2: If | w v,r − w v,r +1 | < C ∗ δ k − (notice the implied existence of w v,r +1 ) then v existsby (V2), so it must be that | v − v | ≥ C ∗ δ k − . Set Γ Rk,v = B [ k, v, v ] .This completes the description of Γ Rk,v . We define the left half, Γ Lk,v , of Γ k,v symmetrically. Let Γ Flat k := S v ∈ V Flat k Γ k,v . If V Non-flat k = ∅ , set Γ k = Γ Flat k ∪ k − [ j = k [ B [ j,w ′ ,w ′′ ] ⊂ Γ j B [ j, w ′ , w ′′ ]; the construction at stage k is complete. Otherwise, we will construct Γ k,v for v ∈ V Non-flat k . We willuse these locally defined sets to define Γ Non-flat k which will then be appended to Γ Flat k . The resultingset will be Γ k Case N Construction. Fix v ∈ V Non-flat k . We first define Γ k,v in terms of a graph. Let E k,v be the setof all edges [ v ′ , v ′′ ] such that [ v ′ , v ′′ ] is an edge in Γ Flat k or in S k or B [ k, v ′ , v ′′ ] is a bridge in Γ Flat k , andeither v ′ or v ′′ is in B ( v, C ∗ δ k − ) . Let V k,v be the set of vertices in V k ∩ B ( v, C ∗ δ k − ) togetherwith any additional endpoints of edges in E k,v . Let G k,v be the graph with edges set E k,v and vertexset V k,v . If G k,v is connected then we let Γ k,v be the set with edges [ v ′ , v ′′ ] or bridges B [ k, v ′ , v ′′ ] such that [ v ′ , v ′′ ] ∈ E k,v . Otherwise, label the connected components of G k,v : G (1) k,v , ..., G ( n ) k,v . Eachconnected component contains at least one non-flat vertex, say v i for G ( i ) k,v . Add edge [ v i , v i +1 ] to anew edge set, E ′ k,v , for ≤ i ≤ n − . Then redefine G k,v to be the graph with edge set E k,v S E ′ k,v and vertex set V k,v .We now consider the global graph G ′ k with edge set E ′ k = S v ∈ V Non-flat k E ′ k,v and vertex set V k = S v ∈ V Non-flat k V k,v . If G ′ k contains cycles, we remove edges from E ′ k one-by-one until no cycles remain. ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 27 The resulting graph G ′ k is a union of trees such that any two vertices which where originally con-nected are still connected. We define Γ Non-flat k to be the set with edges [ v ′ , v ′′ ] or bridges B [ k, v ′ , v ′′ ] such that [ v ′ , v ′′ ] ∈ (cid:0)S v ∈ V k E k,v (cid:1) S E ′ k and vertex set S v ∈ V k V k,v . When | v ′ − v ′′ | < C ∗ δ k − ,we add the new edge [ v ′ , v ′′ ] to Edge ( k ) (this includes all edges from S k ) and when | v ′ − v ′′ | ≥ C ∗ δ k − we add the new bridge B [ k, v ′ , v ′′ ] to Bridge Non-flat ( k ) . Finally, we set Γ k = Γ Flat k ∪ Γ Non-flat k ∪ k − [ j = k [ B [ j,w ′ ,w ′′ ] ⊂ Γ j B [ j, w ′ , w ′′ ] . Connectedness. We will now prove that Γ k is connected for each k ≥ k . Again, we relyheavily on the proof of connectedness in [BS17]. We remark that the use of the exponent k − rather than k in the bound distinguishing between edges and bridge for the case α k,v < ǫ followsfrom the use property (V2) in the proof of connectedness. We provide details of the proof tohighlight where the smaller exponent is needed.For k ≥ k , every point x ∈ Γ k is connected to V k in Γ k because x belongs to an edge [ v ′ , v ′′ ] between vertices v ′ , v ′′ ∈ V k or to some bridge B [ j, u ′ , u ′′ ] between vertices u ′ , u ′′ ∈ V j for some k ≤ j ≤ k . Thus, as in [BS17], to prove that Γ k is a connected set, it suffices to prove that everypair of vertices in V k is connected in Γ k . We use a double induction scheme as in [BS17, Section8.3] to prove that if for any k ≥ k + 1 , if Γ k − is connected then Γ k is connected.Our initial induction is on k . For the base case, generation k , we consider two cases. Firstsuppose that V Flat k = ∅ or V Non-flat k contains a semi-flat vertex. Then recall there exists a linearordering on all points in V k , v − l , ...v , ...v m , and Γ k is constructed by connected by adding anedge [ v i , v i +1 ] for − l ≤ i ≤ m − . In particular, for s > r , v r is connected to v s by the sequenceof edges [ v r , v r +1 ] , ..., [ v s − , v s ] . Suppose instead that V Flat k = ∅ and V Non-flat k does not contain anysemi-flat vertex. Then Γ k is defined to be a connected graph on the vertices in V k so the resultholds trivially.Now suppose that Γ k − is connected for some k ≥ k + 1 . Note that it follows trivially fromconstruction in both the flat case and the non-flat case that if v ′ , v ′′ ∈ V k and | v ′ − v ′′ | < C ∗ δ k − ,then v ′ and v ′′ are connected in Γ k . Let x and y be arbitrary vertices in V k and let w x , w y ∈ V k − denote vertices that are closest to x and y respectively. Since V k − is connected, w x and w y can bejoined in Γ k − by a tour of p + 1 vertices in V k +1 , say, w = w x , w , w , ..., w p = w y where each pair w i , w i +1 of consecutive vertices is connected in Γ k − by an edge [ w i , w i +1 ] or bya bridge B [ j, u ′ , u ′′ ] for some k ≤ j ≤ k − and u ′ , u ′′ ∈ V j with the property that w i ∈ e [ j, u ] and w i +1 ∈ e [ j, u ] . Set v = x . By (V3) and the choice of w to be a closest point to x , we have, | v − w | = | x − w x | < C ∗ δ k .We are now begin our second induction. For any ≤ t ≤ p − there exists a vertex v t ∈ V k such that | v t − w t | < C ∗ δ k − by (V2). Assume that v and v t are connected in Γ k . If t ≤ p − ,choose the vertex v t +1 to be any vertex in V k satisfying | v t +1 − w t +1 | < C ∗ δ k − ; such vertex existsby (V2). Otherwise, if t = p − , set v t +1 = v p = y , which of course satisfies | v t +1 − w t +1 | = | y − w y | < C ∗ δ k < C ∗ δ k − by (V3) and by choice of w y as the closest vertex in V k − . We willshow that v t and v t +1 are connected in Γ k in order to conclude that v and v t +1 are connected in Γ k . We consider two cases:(1) w t and w t +1 are connected by a bridge.(2) w t and w t +1 are connected by an edge. First suppose that w t and w t +1 are connected by a bridge B [ j, u ′ , u ′′ ] for u ′ , u ′′ ∈ V j where k ≤ j ≤ k − . In particular, suppose w t ∈ e [ j, u ′ ] and w t +1 ∈ e [ j, u ′′ ] . Let z ′ denote the point in V k ∩ e [ j, u ′ ] and z ′′ denote the point in V k ∩ E [ j, u ′′ ] . Since z ′ , z ′′ ∈ B [ j, u ′ , u ′′ ] ⊂ Γ k and bridgesare connected subsets of Γ k , z ′ and z ′′ are connected in Γ k . Now by definition of extension in termsof nearest points and by (V2), | z ′ − w t | < C ∗ δ k − . Thus | v t − z ′ | ≤ | v t − w t | + | w t − z ′ | < C ∗ δ k − < C ∗ δ k − . An analogous estimation show that | v t +1 − z ′′ | < C ∗ δ k − . It follows that v t is connected to z ′ and v t +1 is connected to z ′′ so v t is connected to v t +1 in Γ k .Secondly, suppose that [ w t , w t +1 ] is an edge in Γ k − . By definition of edge, we know that | w t − w t +1 | < C ∗ δ k − . Hence | v t − v t +1 | ≤ | v t − w t | + | w t − w t +1 | + | w t +1 − v t +1 | ≤ C ∗ δ k − + 30 C ∗ δ k − < C ∗ δ k − . To conclude the proof of the connectedness, we consider two cases depending on whether α k,v t < ǫ or α k,v t ≥ ǫ . When α k,v t ≥ ǫ , we are in the Case N construction of Γ k . In this case, we defined Γ k,v t to be a connected graph with vertices in B ( v t , C ∗ δ k − ) so, in particular, v t is connected to v t +1 in Γ k,v t . The reduction of edges to construct Γ Non-flat k did not affect connectedness.On the other hand, when α k,v t ≤ ǫ the vertices in V k ∩ B ( v t , C ∗ δ k − ) can be arranged linearlyaccording to the relative ordering under orthogonal projection onto ℓ k,v . We label the vertices in V k ∩ B ( v t , C ∗ δ k − ) lying between v t and v t +1 according to that ordering, z = v t , z , ..., z q = v t +1 . Since (1 + 3 ǫ )32 < , Lemma 6.2 guarantees that v t , v t +1 ∈ B ( z i , C ∗ δ k − ) for all ≤ i ≤ q .Suppose that α k,z i < ǫ for all ≤ i ≤ q . Since Γ k − contains the edge [ w t , w t +1 ] , the set Γ k,z i contains either a bridge B [ k, z i , z i +1 ] or and edge [ z i , z i +1 ] for each ≤ i ≤ q − depending onwhether z i is terminal or not terminal to z i +1 . (We emphasize that Case F T1 does not occur heresince w t +1 exists.) Hence z i and z i +1 are connected for all ≤ i ≤ q − . By concatenating paths,we see that v t = z and v t +1 = z q are connected in Γ k as well. Suppose instead that there existssome i such that α k,z i ≥ ǫ . Then again by the Case F construction of Γ k,z i as a connected graph, z is connected to z q , i.e. v t is connected to v t +1 .By induction, v and v t are connected in Γ k for all ≤ t ≤ p . In particular, we note that x = v and y = v p are connected in Γ . Since x and y are arbitrary in V k , it follows that V k is connected in Γ k . Again by induction, Γ k is connected for all k > k .6.3. Length estimates. The goal of this section is to find length estimates for Γ k , k ≥ k whichthen provide the desired bound for the length of the limiting curve Γ . We first bound the lengthof Γ k either in terms of C ∗ δ k or by the sum over α k ,v over v ∈ V k . We then bound H (Γ k ) by H (Γ k − ) + C P v ∈ V k α k,v δ k for all k ≥ k + 1 where C is independent of k . We follow theoutline of [BS17] and indicate changes required, particularly near vertices v ∈ V Non-flat ( k ) and forthe Case F-NT construction. Before we begin the estimates, we introduce the notion of “phantomlength.”6.3.1. Phantom length. As in [BS17], we will use phantom length to overcome the challenge ofterminal vertices where the old curve does not span the new curve. We define phantom lengthanalogously to the definition in [BS17, Section 9.1]; we provide the following exposition in orderto introduce terminology that will be used in later estimates. ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 29 To begin we establish notation to that will allow us to refer to specific vertices in the extensionsof a bridge. For each extension e [ k, v ] , say e [ k, v ] = ∞ [ i =1 [ v i , v i +1 ] define the corresponding extension index set I [ k, v ] by I [ k, v ] = { ( k + i, v i ) , i ≥ } . Then for each bridge, B [ k, v ′ , v ′′ ] , we define the corresponding bridge index set I [ k, v ′ , v ′′ ] by I [ k, v ′ , v ′′ ] = I [ k, v ′ ] ∪ I [ k, v ′′ ] . For all generations k ≥ k and for all vertices v ∈ V k , we define that phantom length p k,v :=3 C ∗ δ k − . In particular, for a B [ k, v ′ , v ′′ ] between vertices v ′ , v ′′ ∈ V k the totality p k,v ′ ,v ′′ of phantomlength associated to the index set is p k,v ′ ,v ′′ := 3 C ∗ ∞ X i =0 δ k + i − + 3 C ∗ ∞ X j =0 δ k + j − < C ∗ δ k − We track phantom length in pairs ( k, v ) so that we can record both the location and length of thephantom length. We initialize Phantom ( k ) to bePhantom ( k ) := { ( k , v ) : v ∈ V k } . Now suppose that Phantom ( k ) , . . . , Phantom ( k − have been defined for each k ≥ k + 1 sothat Phantom ( k − satisfies the following two properties:(1) Bridge Property: If a bridge B [ k − , w ′ , w ′′ ] is included in Γ k − then Phantom ( k − contains I [ k − , w ′ , w ′′ ] .(2) Terminal Vertex Property: Let w ∈ V k − be a terminal vertex, and let ℓ be a line such that dist( y, ℓ ) < ǫδ k − for all y ∈ V k − ∩ B ( w, C ∗ δ k − ) . Arrange V k − ∩ B ( w, C ∗ δ k − ) linearly with respect to the orthogonal projection π ℓ onto ℓ . If there is no vertex to the “left”of w or to the “right” of w , then ( k − , w ) ∈ Phantom ( k − .Note that Phantom ( k ) satisfies the Bridge Property trivially since no bridges are added during theinitial stage of construction and satisfies the Terminal Vertex Property trivially since Phantom ( k ) includes ( k , v ) for every v ∈ V k . We use Phantom ( k − as a basis for defining Phantom ( k ) . Inparticular, we initialize Phantom ( k ) by setting it to Phantom ( k − . Next, we delete all pairs ofthe form ( k − , w ) or ( k, ˜ v ) that appear in Phantom ( k − from Phantom ( k ) . Finally, for eachvertex v ∈ V k , we include additional pairs in Phantom ( k ) according to the following rules: Case F-NT: If α k,v < ǫ and Γ Rk,v and Γ Lk,v are both defined using Case F-NT then ( k, v ) does notgenerate any new phantom length. Case F-T1: If α k,v < ǫ and either Γ Rk,v or Γ Lk,v is defined by Case F-T1 then include ( k, v ) ∈ Phantom ( k ) . Case F-T2: Suppose α k,v < ǫ and either Γ Rk,v or Γ Lk,v is defined using Case F-T2 . When Γ Rk,v isdefined by Case F-T2 , include I [ k, v, v ] as a subset of Phantom ( k ) . When Γ Lk,v is defined by CaseF-T2 , include I [ k, v − , v ] as a subset of Phantom ( k ) . In particular, in either case ( k, v ) is includedin Phantom ( k ) . Case N: If α k,v ≥ ǫ , include ( k, v ′ ) in Phantom ( k ) for all vertices v ′ ∈ V Non-flat k ∩ B ( v, C ∗ δ k − ) .Additionally, include I [ k, v ′ , v ′′ ] as a subset of Phantom ( k ) for every bridge B [ k, v ′ , v ′′ ] in Γ k,v . Clearly, Phantom ( k ) satisfies the bridge property. To check that Phantom ( k ) satisfies that ter-minal vertex property, let v ∈ V k be a terminal vertex, and suppose that we can find a line ℓ suchthat dist( y, ℓ ) < ǫδ k for all y ∈ V k ∩ B ( v, C ∗ δ k − ) . Identify ℓ with R n and arrange V k ∩ B ( v, C ∗ δ k − ) linearly with respect to the orthogonal pro-jection π ℓ onto ℓ . Assume there is no vertex v ′ ∈ V k ∩ B ( v, C ∗ δ k − ) to the “left” of v orto the “right” of v with respect to the ordering under π ℓ . If α k,v ≥ ǫ , then ( k, ˜ v ) was includedin Phantom ( k ) for every ˜ v ∈ V Non-flat k ∩ B ( v, C ∗ δ k − ) . In particular, ( k, v ) is in Phantom ( k ) .Otherwise α k,v < ǫ , so V k ∩ B ( v, C ∗ δ k − ) is also linearly ordered with respect to orthogonalprojection onto ℓ k,v . By Lemma 6.2, the orderings agree modulo the choice of orientation for thelines. The assumption that there is no vertex v ′ ∈ V k ∩ B ( v, C ∗ δ k − ) to the “left” or to the“right” translates to the statement that Γ Lk,v or Γ Rk,v is defined by Case F-T1 or Case F-T2 , so ( k, v ) was included in Phantom ( k ) . Therefore, Phantom ( k ) satisfies the terminal vertex property.6.3.2. Cores of Bridges. For each bridge B [ k, v ′ , v ′′ ] ∈ Bridge Flat ( k ) between vertices v ′ , v ′′ ∈ V k ,we define the core C [ k, v ′ , v ′′ ] of the bridge to be C [ k, v ′ , v ′′ ] := 910 [ v ′ , v ′′ ] i.e., C [ k, v ′ , v ′′ ] is the interval of length of the length of [ v ′ , v ′′ ] that is concentric to [ v ′ , v ′′ ] .Recall that H ( B [ k, v ′ , v ′′ ]) ≥ C ∗ δ k − for every bridge B [ k, v ′ , v ′′ ] ∈ Bridge Flat ( k ) . Thus thecorresponding core also has significant length, H ( C [ l, v ′ , v ′′ ]) ≥ C ∗ δ k − . Cores in Cores Flat ( k ) are disjoint; see [BS17, Section 9.2]. We emphasize that here we only definethe cores for bridges in Bridge Flat ( k ) Proof of Theorem C. To establish Theorem C, it suffices to prove that(13) X [ v ′ ,v ′′ ] ∈ Edge ( k ) H ([ v ′ , v ′′ ]) + X ( j,u ) ∈ Phantom ( k ) p j,u ≤ Cδ k + X v ∈ V α k ,v δ k , and then that for all k ≥ k + 1 X [ v ′ ,v ′′ ] ∈ Edges ( k ) H ([ v ′ , v ′′ ]) + X B [ k,v ′ ,v ′′ ] ∈ Bridge ( k ) H ( B [ k, v ′ , v ′′ ]) + X ( j,u ) ∈ Phantom ( k ) p j,u ≤ X [ w ′ ,w ′′ ] ∈ Edges ( k − H ([ w ′ , w ′′ ]) + X ( j,u ) ∈ Phantom ( k − p j,u + C X v ∈ V k α k,v δ k + 2527 X C [ k,v ′ ,v ′′ ] ∈ Cores Flat ( k ) H ([ k, v ′ , v ′′ ] . ) , (14)where C denotes a constant depending only on C ∗ and δ . To see that establishing these bounds issufficient, iterate (14) k − k times and then apply (13). See [BS17, Section 9.3] for details. ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 31 Preliminary Observation. We begin with a preliminary observation about the lengths ofedges and bridges that will be used in the proofs of the two in equalities. Recall that an edge [ v ′ , v ′′ ] in the curves Γ k , Γ k +1 , ... is included for some v ′ , v ′′ ∈ V k only if | v ′ − v ′′ | < C ∗ δ k − ,while a bridge B [ k, v ′ , v ′′ ] ∈ Bridge ( k ) is included for some v ′ , v ′′ ∈ V k only if C ∗ δ k − ≤| v − v ′ | ≤ C ∗ δ k − . Furthermore, the lengths of the extensions are controlled by (V2): For all k ≥ k and v ∈ V k , H ( e [ k, v ]) ≤ C ∗ δ k . Thus, if B [ k, v ′ , v ′′ ] ∈ Bridge ( k ) then H ( B [ k, v ′ , v ′′ ]) ≤ H ( e [ k, v ′ ]) + H ([ v ′ , v ′′ ]) + H ( e [ k, v ′′ ]) ≤ C ∗ δ k − + H ([ v ′ , v ′′ ]) ≤ δ + 3030 H ([ v ′ , v ′′ ]) < H ([ v ′ , v ′′ ]) . Length Estimates for Base Case k . Recall that there are no bridges added during the con-struction of Γ k Since H ([ v ′ , v ′′ ])) ≤ C ∗ δ k − for every [ v ′ , v ′′ ] ∈ Edge ( k ) ,(15) X [ v ′ ,v ′′ ] ∈ Edge ( k ) H ([ v ′ , v ′′ ]) ≤ V k C ∗ δ k − . Additionally(16) X ( j,u ) ∈ Phantom ( k ) p j,u = X v ∈ V k p k ,v ≤ V k C ∗ δ k − . Now we consider two cases. Suppose first that V Flat k = ∅ . Fix v such that α k ,v < ǫ , andconsider the corresponding approximating line ℓ k ,v . For any v , v ∈ V k , consider π ( v ) , π ( v ) ,their respective projections onto ℓ k ,v . We have | π ( v ) − π ( v ) | ≥ | v − v | − dist( v , ℓ k ,v ) − dist( v , ℓ k ,v ) ≥ C ∗ δ k − C ∗ ǫδ k > (1 − ǫ ) C ∗ δ k . Since π ( v i ) ∈ B ( v , C ∗ δ k − ) we see that V . C ∗ ,δ . In particular, X [ v ′ ,v ′′ ] ∈ Edge ( k ) H ([ v ′ , v ′′ ]) + X ( j,u ) ∈ Phantom ( k ) p j,u . C ∗ ,δ δ k . Alternatively, suppose that V Flat ( k ) = ∅ . Then for each added line segment in Γ k , the length ofthe line segment is charged against the large α k ,v value for a unique v ∈ V Non-flat ( k ) . We alsocharge the phantom length assigned at each vertex v to the large α k ,v value. That is, X [ v ′ ,v ′′ ] ∈ Edge ( k ) H ([ v ′ , v ′′ ]) + X ( j,u ) ∈ Phantom ( k ) p j,u . C ∗ ,δ X v ∈ V k α k ,v δ k , Combining these two estimates we conclude that inequality (13) holds.6.3.6. Length Estimates for k > k . We are now ready to work on the proof of (14). Note thatedges and bridges forming the curve Γ k and “new” phantom length may appear in the local portionof Γ k near v , namely Γ k,v , for several vertices v ∈ V k but only need to be accounted for once eachin order to estimate the left hand side of (14). We will present the length estimates for Case N construction first and then we will present estimates for Case F construction. We will refer readersto [BS17, Section 9.5] for some details of the Case F construction estimates. Case N : Here we will pay of edges or bridges in Γ k \ Γ Flat k as well as well as any parts of edgesin B ( v, C ∗ δ k − ) for v ∈ V Non-flat k that were added during a Case F stage of construction. We willcharge the length to the large α k,v value corresponding to vertices v ∈ V k . By Lemma 6.2, for a semi-flat vertex v ∈ V Non-flat k , the sum of the length of edges in S k associated to vertex v cannotexceed ǫ ) C ∗ δ k − < C ∗ δ k − ≤ (cid:18) C ∗ ǫ (cid:19) α k,v δ k − . Additionally, since G ′ k is a union of disjoint trees, each edge [ v, v ′ ] in G ′ k can be assigned uniquelyto a vertex, say v ∈ V Non-flat k . Then since α k,v ≥ ǫ , if the corresponding edge [ v, v ′ ] was added to Γ Non-flat k then H ([ v, v ′ ]) ≤ C ∗ ǫ − α k,v ′ δ k − . If instead the corresponding bridge B ([ v, v ′ , k ]) wasadded in the construction of Γ Non-flat k then H ( B [ v, v ′ , k ]) ≤ (cid:18) (cid:19) C ∗ δ k − , so H ( B [ v, v ′ , k ]) ≤ C ∗ ǫ − α k,v ′ δ k − . Finally, the length of parts of edges in B ( v, C ∗ δ k − ) added during a Case F stage of construction is at most (1+3 ǫ )4 C ∗ δ k ≤ C ∗ δ k − . Let Edge Non-flat ( k ) denote the set of edges in E ( k ) such that [ v ′ , v ′′ ] ∈ E ′ k or [ v ′ , v ′′ ] ∈ S k . Then X [ v ′ ,v ′′ ] ∈ Edge Non-flat ( k ) H ([ v ′ , v ′′ ]) + X B ([ v ′ ,v ′′ ,k ]) ∈ Bridge Non-flat ( k ) H ( B [ v ′ , v ′′ , k ]) + 5 C ∗ δ k − . C ∗ ,δ X v ∈ V Non-flat k α k,v δ k . (17)We emphasize that here we rely on the fact that we constructed G ′ k to be the union of trees, so wecan charge each edge of E ′ k to a unique vertex v ∈ V Non-flat k . We also bound the phantom length asfollows X v ∈ V Non-flat k p k,v + X B [ v ′ ,v ′′ ,k ] ∈ Bridge Non-flat ( k,v ) p k,v ′ ,v ′′ ≤ X v ′ ∈ V Non-flat k C ∗ δ k − + X Bridge [ v ′ ,v ′′ ,k ] ∈ Bridge Non-flat ( k,v ) C ∗ δ k − . C ∗ ,δ X v ∈ V Non-flat k α k,v δ k . Case F T1 : This estimate follows identically to as in Section 9.5 of [BS17]. In particular, p k,v + X [ v ′ ,v ′′ ] ∈ Edges ( k ) H ([ v ′ , v ] ∩ B ( v, C ∗ δ k )) ≤ p k − ,w v,r . Case F T2 : Suppose v is terminal to the right with alternative T2. Recall that in this step weneed to a add a bridge in Bridge Flat ( k ) . Write v ∈ V k and w v,r , w v,r +1 ∈ V k − for the verticesappearing in the definition of Γ Rk,v . In this case, we will pay for p k,v,v , the length of the bridge B [ k, v, v ] and the length of the edges in Γ k ∩ B ( v, C ∗ δ k − ) . We will also pay for the length in Γ k ∩ B ( v , C ∗ δ k − ) if we have not already done so. As previously noted, H ( B [ k, v, v ]) ≤ C ∗ δ k + H ([ v ′ , v ′′ ]) , Since | v − w v,r | < C ∗ δ k and | v − w v,r +1 | < C ∗ δ k − , it follows that H ( B [ k, v, v ]) ≤ C ∗ δ k − + H ([ v, v ]) ≤ C ∗ δ k − + H ([ w v,r , w v,r +1 ]) . ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 33 Note that if w v,r / ∈ V Flat k − then w v,r is a semi-flat vertex. In either case, the edge [ w v,r , w w,r +1 ] is in Γ k − . Additionally, the totality of phantom length associated with vertices in B [ k, v, v ] is C ∗ δ k − . Unlike in [BS17, Section 9.5], we cannot assume α k,v < ǫ . However, if α k,v ≥ ǫ thenwe have already paid for the length of Γ k ∩ B ( v , C ∗ δ k ) . In this case, H ( B [ k, v, v ]) + p k,v,v + X [ v ′ ,v ′′ ] ∈ Edges ( k ) H ([ v ′ , v ′′ ] ∩ B ( v, C ∗ δ k ))) ≤ H ([ w v,r , w v,r +1 ]) + 23 C ∗ δ k − ≤ H ([ w v,r , w v,r +1 ]) + 2327 H ( C [ k, v, v ]) where [ w v,r , w v,r +1 ] ∈ Edges ( k − and C [ k, v, v ] ∈ Cores Flat ( k ) . Otherwise, α k,v < ǫ. In thiscase, the total length of parts of edges in Γ k ∩ B ( v, C ∗ δ k − ) ∪ B ( v , C ∗ δ k − ) which has not yetbeen paid for does not exceed C ∗ δ k − by Lemma 6.2. Altogether these estimates sum to give thebounds H ( B [ k, v, v ]) + p k,v,v + X [ v ′ ,v ′′ ] ∈ Edges ( k ) H ([ v ′ , v ′′ ] ∩ B ( v, C ∗ δ k ) ∪ B ( v , C ∗ δ k ))) ≤ H ([ w v,r , w v,r +1 ]) + 25 C ∗ δ k − ≤ H ([ w v,r , w v,r +1 ]) + 2527 H ( C [ k, v, v ]) where [ w v,r , w v,r +1 ] ∈ Edges ( k − and C [ k, v, v ] ∈ Cores Flat ( k ) . Case F NT : Let [ v ′ , v ′′ ] be an edge between vertices v ′ , v ′′ ∈ V k which are not yet wholly paidfor. Then there exists a vertex v ∈ V k such that | v − v ′ | < C ∗ δ k − , | v − v ′′ | < C ∗ δ k − , | v ′ − v ′′ | < C ∗ δ k − , and v ′ is immediately to the left (or to the right) of v ′′ relative to the orderdefined by ℓ k,v . Let [ u ′ , u ′′ ] be the largest closed subinterval of [ v ′ , v ′′ ] such that u ′ and u ′′ lie adistance at least C ∗ δ k − away from Case F-T1 and Case F-T2 vertices as well as vertices inV Non-flat k . Note that we already paid for the length within distance C ∗ δ k − of these three types ofvertices. Applying Lemma 6.2, H ([ u ′ , u ′′ ]) ≤ (1 + 3 α k,v ′ ) H ([ π k,v ′ ( u ′ ) , π k,v ′ ( u ′′ )])= H ([ π k,v ′ ( u ′ ) , π k,v ′ ( u ′′ )]) + 90 C ∗ α k,v δ k − . Without loss of generality, suppose that u ′ lies to the left of u ′′ relative to the order of their re-spective projections on ℓ k,v ′ . Let z ′ denote the first vertex in V k ∩ B ( v ′ , C ∗ δ k − ) to the leftof u ′ , relative to the order of their projection onto ℓ k,v , such that π k,v ′ ( z ′ ) < π k,v ( u ′ ) − C ∗ δ k .Analogously, let z ′′ denote the first vertex in V k ∩ B ( v, C ∗ δ k − ) to the right of u ′′ , such that π k,v ( u ′′ ) + C ∗ δ k < π k,v ( z ′′ ) . The vertex z ′ as described above always exists since, if z ′ = v ′ then | v ′ − u ′ | ≤ C ∗ δ k . Thus v ′ must be a Case F-NT vertex; a similarly conclusion holds for v ′′ . Thisimplies that | z ′ − v ′ | < C ∗ δ k − and | z ′′ − v ′′ | < C ∗ δ k − . By (V3), we can find w ′ , w ′′ ∈ V k − such that | w ′ − z ′ | < C ∗ δ k and | w ′′ − z ′′ | < C ∗ δ k . By choice of w ′ and w ′′ , π k,v ′ ( w ′ ) < π k,v ′ ( u ′ ) < π k,v ′′ ( u ′′ ) < π k,v ′ ( w ′′ ) . We claim that there exists a sequence of edges in Γ k − connecting w ′ to w ′′ such that the edges arecontained in an C ∗ δ k ǫ - neighborhood of ℓ k,v . To see that this claim is true, recall that by (V3) thereare y ′ , y ′′ ∈ V k − such that | y ′ − v ′ | < C ∗ δ k and | y ′′ − v ′′ | < C ∗ δ k − . If α k − ,y ′ < ǫ , then thereexists an ordering on the points in V k − ∩ B ( y ′ , C ∗ δ k − ) given by projection onto ℓ k − ,y ′ . In this case | w ′ − y ′ | ≤ | w ′ − z ′ | + | z ′ − v ′ | + | v ′ − y ′ | < C ∗ δ k − , so a sequence of edges between w ′ and y ′ was added during a Case F-NT stage of construction of Γ k − . A similar estimation showsthat | y ′ − y ′′ | < C ∗ δ k − so there is sequence of edges between y ′ and y ′′ . If instead α k − ,y ′ ≥ ǫ ,then y ′ is a semi-flat vertex, and, by Lemma 6.2, the same sequence of edges was added to Γ k − in the Case N construction. Now y ′′ satisfies α k − ,y ′′ < ǫ or y ′′ is a semi-flat vertex. In eithercase, since | y ′′ − w ′′ | < C ∗ δ k − , there is a sequence of edges connecting y ′′ to w ′′ in Γ k − . Weemphasize that since | w ′ − v | < C ∗ δ k − and | w ′′ − v | < C ∗ δ k − , the edges added during theconstruction of Γ k − agree with ordering of points according to projection onto ℓ k,v . Furthermore,since all x ∈ V k − ∩ B ( v, C ∗ δ k − ) are distance less than C ∗ δ k ǫ away from ℓ k,v , the portion of Γ k − between w ′ and w ′′ is distance less than C ∗ δ k ǫ from ℓ k,v .We can pay for H ([ π k,v ′ ( u ′ ) , π k,v ′ ( u ′′ )]) using the portion of edges in the curve Γ k − ∩ B ( v, C ∗ δ k − ) that lies over the segment [ π k,v ( u ′ ) , π k,v ( u ′′ )] . Thus, H ([ u ′ , u ′′ ]) ≤ H ( E k − ( v ) ∩ π − k,v ([ π k,v ( u ′ ) , π k,v ( u ′′ )])) + 90 C ∗ α k,v δ k − where E k − ( v ) denotes the union of edges in Γ k − between the vertices in V k − ∩ B ( v, C ∗ δ k − ) .It remains to estimate the overlap of the sets of the form S k,v [ u ′ , u ′′ ] := E k − ( v ) ∩ π − k,v ([ π k,v ( u ′ ) , π k,v ( u ′′ )]) Since S k,v ′ ([ u ′ , u ′′ ]) ⊂ S k,v ′ ([ v ′ , v ′′ ]) , it suffices to estimate the length of the overlap of sets S k,v ′ [ v ′ , v ′′ ] . Suppose that v , v , v are consecutive vertices in V k ∩ B ( v (1) , C ∗ δ k − ) such thatportions of edges [ v , v ] and [ v , v ] are being paid for in this Case F-NT stage. Suppose that that [ v , v ] was added during the construction of Γ k,v (1) and [ v , v ] was added during the constructionof Γ k,v (2) where v (1) , v (2) ∈ V Flat k are both non-terminal. We will show that H ( S k,v (1) [ v , v ] ∩ S k,v (2) [ v , v ]) < α δ k − where α = max { α k,v (1) , α k,v (2) } . To start, let ℓ be a line which is parallel to ℓ k,v (1) but passesthrough v , and similarly let ℓ be a line which is parallel to ℓ k,v (2) and passes through v . Let π i denote orthogonal projection onto ℓ i and let N i denote the closed tubular neighborhood of ℓ i ofradius αδ k . Also, let E k − ( v (1) , v (2) ) := E k − ( v (1) ) ∩ E k − ( v (2) ) . Then S k,v (1) [ v , v ] ∩ S k,v (2) [ v , v ] ⊂ E k − ( v (1) , v (2) ) ∩ π − ([ π ( v ) , π i ( v )]) ∩ N ∩ π − ([ π ( v ) , π ( v )]) ∩ N =: E k − ( v (1) , v (2) ) ∩ S. The remainder of the overlap estimate follows identically as in [BS17, Section 9.5]. Now wecombine all the estimates above to conclude (14).7. G RAPH RECTIFIABLE MEASURES In this section we will prove Theorem D. Throughout H denotes a finite or infinite dimensionalHilbert space. Recall that we define the good cone at x with respect to V and α by C G ( x, V, α ) := { y ∈ H : dist( y − x, V ) ≤ α | x − y |} , and the bad cone at x with respect to V and α by C B ( x, V, α ) := H \ C G ( x, V, α ) . We begin by collecting some geometric results that will be useful in the proof of Theorem D.The first result can be found in [Mat95]. We present the proof, with slight modifications, in theappendix to highlight some important consequences. ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 35 Theorem 7.1 (Geometric Lemma) . Let F ⊂ H , let V be an m -dimensional linear plane in H , andlet α ∈ (0 , . If F \ C G ( x, V, α ) = ∅ for all x ∈ F then F is contained in an m -Lipschitz graphs. In particular, F ⊂ Γ where Γ is a Lipschitz graphwith respect to V and the Lipschitz constant corresponding to Γ is at most / (1 − α ) / . Since we are interested in measure-theoretic results up to sets of measure zero we provide acorollary to Theorem 7.1. Corollary 7.1. Let µ be a Radon measure on H , V be an m -dimensional linear plane in H , α ∈ (0 , , and < r < ∞ . If for µ -a.e. x ∈ H (18) µ ( C B ( x, r, V, α )) = 0 then µ is carried by m -Lipschitz graphs.Proof. Let F denote the set of x ∈ H that satisfy (18). We may assume F ⊂ B (0 , r/ ; otherwisewe may write F as a union of countably many sufficiently small sets and show that each one is m -graph rectifiable. Let { x i } be a countable dense subset of F . It follows from (18) and thecontainment F ⊂ B (0 , r/ that for each x i there exists F i ⊂ F such that F i ∩ C B ( x i , r, V, α ) = F i ∩ C B ( x i , V, α ) = ∅ and µ ( F \ F i ) = 0 . Define F ′ := T ∞ i =1 F i . Then µ ( F \ F ′ ) = µ F \ ∞ \ i =1 F i ! = µ ∞ [ i =1 F \ F i ! ≤ ∞ X i =1 µ ( F \ F i ) = 0 . We claim that F ′ ∩ C B ( x, V, α ) = ∅ for every x ∈ F ′ . Fix x ∈ F ′ , and let y ∈ C B ( x, V, α ) .By definition of bad cone we have that dist( y − x, V ) > α | y − x | . Now let ǫ > such that dist( y − x, V ) ≥ α ( | y − x | + ǫ ) . Recalling that < α < , choose x i such that | x i − x | < αǫ/ <ǫ/ . Then dist( y − x i , V ) ≥ dist( y − x, V ) − | x − x i |≥ α ( | y − x | + ǫ ) − α ( ǫ/ α ( | y − x | + ǫ/ > α ( | y − x | + | x i − x | ) ≥ α ( | y − x i | ) . In particular, we conclude that y ∈ C B ( x i , V, α ) . Since F i ∩ C B ( x i , V, α ) = ∅ , it must be thatcase that y / ∈ F i . It follows that y / ∈ F ′ , and thus F ′ ∩ C B ( x, V, α ) = ∅ for all x ∈ F ′ . By anapplication of Theorem 7.1 we conclude that there exists an m -Lipschitz graph Γ such that F ′ ⊂ Γ ,so µ ( F \ Γ) = 0 . (cid:3) Lemma 7.1. Let x ∈ H , α ∈ (0 , , and V be an m -dimensional linear plane. If y ∈ C B (cid:0) x, V, α + − α (cid:1) then there exists some constant η α depending on at most α and the dimension of the space, n , suchthat B ( y, η α d ) ⊂ C B ( x, V, α ) where d := | x − y | . A proof of Lemma 7.1 can be found in the appendix. With the above results established, we nowprove a lemma that forms the central argument for the proof of the sufficient condition of TheoremD. Lemma 7.2. Let µ be a Radon measure on H . For x ∈ H , V an m -dimensional linear plane, α ∈ (0 , , and parameter K > , let E denote the set of points x ∈ H such that (i) The sequence of functions f r ( x ) := µ ( C B ( x, r, V, α )) µ ( B ( x, r )) converges to uniformly on E , and (ii) there exists r > such that at every x ∈ E , µ ( B ( x, r )) ≤ Kµ ( B ( x, r )) for all r ∈ (0 , r ] . Then E is µ -carried by m -Lipschitz graphs with Lipschitz constants depending on at most K and α .Proof. Fix δ > . By uniform convergence, choose r δ ≤ r such that for all r < r δ and for all x ∈ E ,(19) µ ( C B ( x, r, V, α )) µ ( B ( x, r )) < δ. Fix x ∈ E , and define S := E ∩ C B ( x, r, V, α ) . Assuming the set is non-empty, fix y ∈ S suchthat | x − y | = max y ∈ S | a − y | =: λr for some < λ ≤ . As an application of Lemma 7.1 choose η α such that B ( y , η α λr ) ⊂ C B ( x, r, V, α ) . Let d = log (cid:16) λ +2 η α λ (cid:17) . Then d η α λr = λ + 2 η α λ η α λr = ( λ + 2) r = | x − y | r + 2 r. In particular, for the specified value of d , B ( x, r ) ⊂ B ( y , d η α λr ) . Applying condition (ii) ofthe set E at the point y we see that(20) µ ( C B ( x, r, V, α )) ≥ µ ( B ( x, η α λr )) ≥ K − d µ ( B ( y , d η α λr )) ≥ K − d µ ( B ( x, r )) Combining inequalities (19) and (20), we get the density ratio bounds δ > µ ( C B ( x, r, V, α )) µ ( B ( x, r )) ≥ K − d for all r < r δ . In particular, this implies that d > − log( δ )log K . Equivalently, log (cid:18) λ + 2 η α λ (cid:19) > − log δ log K , so that if δ is chosen to be less than − log K log ( ηα ) then λ < . From this result we conclude that for r < r δ and for all y ∈ S , | x − y | < r . Letting r ↓ we conclude that µ ( E ∩ C B ( x, r δ , V, α )) = 0 .Thus we can apply Corollary 7.1, and we obtain the desired conclusion. (cid:3) With Lemma 7.2 established, we are ready to prove Theorem D. Proof. We first show the sufficient condition holds. To do so, we use a series of countable decom-positions to reduce to a setting in which Lemma 7.2 can be applied. First we may assume that µ is a finite measure, for if µ is not finite then by separability of H we may write H as a countableunion of closed balls of radius . It follows from our definition of pointwise doubling measuresthat µ is finite on each ball in the union. Then the proof proceeds as below by considering therestriction of µ to each ball. ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 37 Choose { V i } ∞ i =1 to be a dense collection of m -dimensional linear planes in H and { α j } ∞ j =1 to bea sequence dense in (0 , . For a fixed α ∈ (0 , and m -dimensional linear plane V , we can find α k > α and V l such that k V l − V k < α k − α . Then we have C B ( x, V, α ) ⊂ C B ( x, V l , α k ) , so ofcourse(21) if lim r ↓ µ ( C B ( x, r, V, α )) µ ( B ( x, r )) = 0 then lim r ↓ µ ( C B ( x, r, V l , α k )) µ ( B ( x, r )) = 0 . Now fix some k and l , and let E k,l := (cid:26) x ∈ H : lim r ↓ µ ( C B ( x, r, V l , α k )) µ ( B ( x, r )) = 0 (cid:27) . By Egorov’s Theorem, choose a measurable subset E k,l,t ⊂ E k,l such that µ ( E k,l \ E k,l,t ) < − t and f k,lr ( x ) := µ ( C B ( x, r, V l , α k )) µ ( B ( x, r )) converges uniformly to zero on E k,l,t . Note that H = S ∞ t =1 S ∞ l =1 S ∞ k =1 E k,l,t so it suffices to showthat E k,l,t is graph rectifiable for fixed k , l , and t . Next, since µ is pointwise doubling, for µ -a.e. x ∈ E k,l,t , there exists K x , N x ∈ N such that µ ( B ( x, r )) ≤ K x µ ( B ( x, r )) for all < r ≤ /N x .Define E K,Nk,l,t = { y ∈ E k,l,t : µ ( B ( y, r )) ≤ Kµ ( B ( y, r )) for all < r ≤ /N } . Then µ (cid:16) E k,l,t \ S ∞ K =1 S ∞ N =1 E K,Nk,l,t (cid:17) = 0 . Finally for µ -a.e. x ∈ E K,Nk,l,t , lim r ↓ µ ( E K,Nk,l,t ∩ B ( x, r )) µ ( B ( x, r )) = 1 . Define E K,Nk,l,t,p = (cid:26) x ∈ E K,Nk,l,t : µ ( E K,Nk,l,t ∩ B ( x, r )) ≥ µ ( B ( x, r )) for all ≤ r ≤ /p (cid:27) , and note that E K,Nk,l,t = S ∞ p =1 E K,Nk,l,t,p . To conclude the proof, apply Lemma 7.2 for some fixed k , l , t , K , N and p .To show the necessary condition, suppose that µ is m -Lipschitz graph rectifiable, and let { Γ i } denote a collection of Lipschitz graphs that carry µ . To each graph Γ i we associate an m -plane V i and a number α i ∈ (0 , such that Γ i is a Lipschitz graph with respect to V i and α i . Let x ∈ H bea µ -density point. Since each graph Γ i is closed, x ∈ Γ i for some i . It follows that lim r ↓ µ ( B ( x, r ) \ Γ i ) µ ( B ( x, r )) = 0 . Furthermore, Γ i ⊂ C G ( x, V i , α i ) , and so C B ( x, r, V i , α i ) ⊂ B ( x, r ) \ Γ i . It follows immediatelythat lim r ↓ µ ( C B ( x, r, V i , α i )) µ ( B ( x, r )) = 0 . This completes the proof of the necessary condition. (cid:3) A PPENDIX A.In this section we collect the proofs of some results that are used a above. The proofs areincluded here for completeness of the exposition. Proof of Lemma 1.2. Let j ≥ k . Let B = B ( x, λ − k ) and B ′ = B ( y, λ − j ) . Suppose that B ∩ B ′ = ∅ . Fix z ∈ B ′ , then dist( z, x ) ≤ dist( z, y ) + dist( y, x ) ≤ λ − k . In particular, B ′ ⊂ B . Furthermore, for z ∈ B , dist( z, y ) ≤ dist( z, x ) + dist( x, y ) ≤ λ − k + λ − j < λ − k , so B ⊂ · j − k B ′ . Let C denote { B ′ i ∈ C µj : B ′ i ∩ B = ∅} . Then µ (2 B ) ≥ µ C [ i =1 B ′ i ! ≥ µ C [ i =1 λ B ′ i ! = C X i =1 µ (cid:18) λ B ′ i (cid:19) ≥ C X i =1 D − ( j − k +3+log( λ )) µ (2 j − k +2 B ′ i ) ≥ C · D − ( j − k +3+log( λ )) µ (2 B ) . (22)This implies D j − k +3+log( λ ) ≥ C . Thus we may take the finite overlap constant to be P µj − k = D j − k +2+log( λ ) . (cid:3) Proof of Lemma 2.1. First note that for each k , Z E k ω ( x ) dν = ∞ X j = k Z E j \ E j +1 ω ( x ) dν. Therefore, ∞ X k =0 c k Z E k ω ( x ) dν = ∞ X k =0 c k ∞ X j = k Z E j \ E j +1 ω ( x ) dν = ∞ X j =0 j X k =1 c k Z E j \ E j +1 ω ( x ) dν = ∞ X j =0 Z E j \ E j +1 j X k =1 c k ω ( x ) dν ≤ ∞ X j =1 Z E j \ E j +1 Cdν = ∞ X j =1 Cµ ( E j \ E j +1 ) ≤ Cµ ( E \ E ) . (cid:3) Proof of Lemma 2.2. Assume P ( E ) < ∞ and that f : E → H is L -Lipschitz. Given ǫ > ,pick η > so that P η ( E ) ≤ P ( E ) + ǫ . Fix < δ ≤ Lη and let { B H ( f ( x i ) , r i ) : i ≤ } be anarbitrary disjoint collection of balls in H centered in f ( E ) such that r i ≤ δ for all i ≥ . Since f is L -Lipschitz, f ( B R ( x i , r i /L )) ⊂ B H ( f ( x i ) , r i ) for all i ≥ . Thus { B R ( x i , r i /L ) : i ≥ } is a disjoint collection of balls in R centered in E such that r i /L ≤ δ/L ≤ η. ECTIFIABILITY OF POINTWISE DOUBLING MEASURES IN HILBERT SPACE 39 Hence ∞ X i =1 (2 r i ) = L ∞ X i =1 (2 r i /L ) ≤ L · P η ( E ) ≤ L ( P ( E ) + ǫ ) . Taking the supremum over all δ packings of f ( E ) we obtain P δ ( f ( E )) ≤ L ( P ( E ) + ǫ ) . Thecorresponding inequality for packing measure P ( E ) follows immediately. (cid:3) Proof of Lemma 2.3. Let E ⊂ A and ǫ > . By definition of packing measure, choose δ > suchthat P δ ( E ) ≤ P ( E ) + ǫ . Using the bounded lower density assumption on µ , for each x ∈ E wecan choose a sequence { r x,i } ∞ i =1 with r x,i ≤ min { δ, r / } with r x,i → as i → ∞ such that foreach i , µ ( B ( x, r x,i )) ≤ λ (2 r x,i ) . Let B = { B ( x, r x,i ) : x ∈ E } where B is a closed ball. By Vitali Covering Theorem (see [Mat95,Theorem 2.2 and Remark 2.3 (b)] and or [Hei01, Theorem 1.6] for results on doubling measuresthat can easily be adapted to the current assumptions), we can choose a subcollection B ′ ⊂ B suchelements of B ′ are disjoint and µ E \ [ B ∈B ′ B ! = 0 . Then µ ( E ) ≤ ∞ X i =1 µ ( B i ) ≤ ∞ X i =1 λ r B i ≤ P δ ( E ) ≤ λ ( P ( E ) + ǫ ) . Let ǫ ↓ to conclude µ ( E ) ≤ λP ( E ) for E ⊂ A . Thus, for A = S ∞ l =1 E l , µ ( A ) ≤ ∞ X l =1 µ ( E l ) ≤ λ ∞ X l =1 P ( E l ) . Hence µ ( A ) ≤ λ P ( A ) . (cid:3) The proof of Lemma 6.1 relies on fundamental properties of excess and Hausdorff distance. Fornonempty sets S, T ⊂ X , the excess, ex ( S, T ) of S over T is defined by ex ( S, T ) := sup s ∈ S inf t ∈ T dist( s, t ) and the Hausdorff distance HD ( S, T ) between S and T is defined by HD ( S, T ) := max { ex ( S, T ) , ex ( T, S ) } . Let CL ( H ) denote the set of nonempty closed subsets of H . Since ( H, | · | ) is a complete metricspace, ( CL ( H ) , HD ) is also a complete metric space. See ([Bee93], Chapter 3) for details. Proof of Lemma 6.1. Let n ≥ , C ∗ > , δ ≤ / and r > . Assume that V , V , V , ... is asequence of nonempty, closed finite subsets of a bounded set B such that each V i satisfies (V2)and (V3). By iterating (V2), we obtain that for any k < j and v k ∈ V k , we can find a sequence of v i ∈ V i , i = k + 1 , ..., j such that | v k − v j | ≤ | v k − v k +1 | + ... + | v j − − v j | < C ∗ δ k r + ... + C ∗ δ j − r ≤ C ∗ δ k . It follows that ex ( V k , V j ) < C ∗ δ k r . Similarly iterating (V3), we obtain that for any k < j , ex ( V j , V k ) < C ∗ δ k r . Thus HD ( V k , V j ) ≤ C ∗ δ k r . In particular this implies that { V k } is a Cauchy sequence of sets.By the completeness of ( CL ( H ) , HD ) , { V k } converges to a closed set V . (cid:3) Proof of Theorem 7.1. Let x ∈ F . Let P V : H → V denote standard projection onto the m -plane V . Suppose that | P V x − P V y | < (1 − α ) / | x − y | . Then y ∈ C B ( x, V, α ) , and by assumption of F this means that y / ∈ F . Thus we may assume that if x, y ∈ F then | P V x − P V y | ≥ (1 − α ) / | x − y | . From this inequality we see that P V | F is one-to-one with Lipschitz inverse f = ( P V | F ) − and Lip ( f ) ≤ (1 − α ) − / . Note that F = f ( P V | F ) . Then there exists a Lipschitz extension ˜ f : V → H so that F ⊂ ˜ f ( V ) . Thus the desired result holds. (cid:3) Proof of Lemma 7.1. To determine the maximum constant η α , we consider a point b ∈ ∂C B (cid:0) x, V, α + − α (cid:1) and determine the distance d ′ from b to C G ( x, V, α ) . Define θ to be the angle between ∂C B (cid:0) x, V, α + − α (cid:1) ,and ∂C B ( x, V, α ) , θ ′ to be the angle between ∂C B ( x, V, α ) and V and θ ′′ to be the angle be-tween ∂C B ( a, V ⊥ , α + − α ) and V . Note that θ = θ ′′ − θ ′ . Some simple calculations show that θ ′′ = cos − (( α + − α ) ′ ) , and θ ′ = cos − (( α ) ′ ) . 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