Two sufficient conditions for rectifiable measures
aa r X i v : . [ m a t h . C A ] J un TWO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES
MATTHEW BADGER AND RAANAN SCHUL
Abstract.
We identify two sufficient conditions for locally finite Borel measures on R n to give full mass to a countable family of Lipschitz images of R m . The first condition,extending a prior result of Pajot, is a sufficient test in terms of L p affine approximabilityfor a locally finite Borel measure µ on R n satisfying the global regularity hypothesislim sup r ↓ µ ( B ( x, r )) /r m < ∞ at µ -a.e. x ∈ R n to be m -rectifiable in the sense above. The second condition is an assumption on thegrowth rate of the 1-density that ensures a locally finite Borel measure µ on R n withlim r ↓ µ ( B ( x, r )) /r = ∞ at µ -a.e. x ∈ R n is 1-rectifiable. Introduction
In the treatise [Fed69] on geometric measure theory, Federer supplies the followinggeneral notion of rectifiability with respect to a measure. Let 1 ≤ m ≤ n − µ be a Borel measure on R n , i.e. a Borel regular outer measure on R n . Then R n iscountably ( µ, m ) rectifiable if there exist countably many Lipschitz maps f i : [0 , m → R n such that µ assigns full measure to the images sets f i ([0 , m ), i.e. µ R n \ ∞ [ i =1 f i ([0 , m ) ! = 0 . When m = 1, each set Γ i = f i ([0 , rectifiable curve . Below we shorten Federer’sterminology, saying that µ is m -rectifiable if R n is countably ( µ, m ) rectifiable.Two well studied subclasses of rectifiable measures are Hausdorff measures on rectifiablesets and absolutely continuous rectifiable measures. Given any Borel measure µ on R n and Borel set E ⊆ R n , define the measure µ E (“ µ restricted to E ”) by the rule µ E ( F ) = µ ( E ∩ F ) for all Borel sets F ⊆ R n . We call a Borel set E ⊆ R n an m -rectifiable set if H m E is an m -rectifiable measure, where H m denotes the m -dimensionalHausdorff measure on R n . One may think of an m -rectifiable set E as an m -rectifiablemeasure by identifying E with the measure H m E . More generally, we say that an m -rectifiable measure µ on R n is absolutely continuous if µ ≪ H m , i.e. µ ( E ) = 0 whenever E ⊂ R n and H m ( E ) = 0. Date : June 29, 2015.2010
Mathematics Subject Classification.
Primary 28A75.
Key words and phrases. rectifiable measure, singular measure, Jones beta number, Hausdorff density,Hausdorff measure.M. Badger was partially supported by an NSF postdoctoral fellowship DMS 1203497. R. Schul waspartially supported by NSF DMS 1361473.
It is a remarkable fact that rectifiable sets and absolutely continuous rectifiable measurescan be identified by the asymptotic behavior of the measures on small balls.
Definition 1.1 (Hausdorff density) . Let B ( x, r ) denote the closed ball in R n with center x ∈ R n and radius r >
0. For each positive integer m ≥
1, let ω m = H m ( B m (0 , R m . For all locally finite Borel measures µ on R n , we definethe lower Hausdorff m -density D m ( µ, · ) and upper Hausdorff m -density D m ( µ, · ) by D m ( µ, x ) := lim inf r → µ ( B ( x, r )) ω m r m ∈ [0 , ∞ ]and D m ( µ, x ) := lim sup r → µ ( B ( x, r )) ω m r m ∈ [0 , ∞ ]for all x ∈ R n . If D m ( µ, x ) = D m ( µ, x ) for some x ∈ R n , then we write D m ( µ, x ) for thecommon value and call D m ( µ, x ) the Hausdorff m -density of µ at x . Theorem 1.2 ([Mat75]) . Let ≤ m ≤ n − . Suppose E ⊂ R n is Borel and µ = H m E is locally finite. Then µ is m -rectifiable if and only if the Hausdorff m -density of µ existsand D m ( µ, x ) = 1 at µ -a.e. x ∈ R n . Theorem 1.3 ([Pre87]) . Let ≤ m ≤ n − . If µ is a locally finite Borel measure on R n ,then µ is m -rectifiable and µ ≪ H m if and only if the Hausdorff m -density of µ exists and < D m ( µ, x ) < ∞ at µ -a.e. x ∈ R n .Remark . For any locally finite Borel measure µ on R n : µ ≪ H m ⇐⇒ D m ( µ, x ) < ∞ at µ -a.e. x ∈ R n ; and, µ is m -rectifiable = ⇒ D m ( µ, x ) > µ -a.e. x ∈ R n . (1.1)See [Mat95, Chapter 6] and [BS15, Lemma 2.7].There are several other characterizations of rectifiable sets and absolutely continuousrectifiable measures (e.g. in terms of projections or tangent measures); see Mattila [Mat95]for a full survey of results through 1993. Further investigations on rectifiable sets andabsolutely continuous rectifiable measures include [Paj96, Paj97, L´eg99, Ler03, Tol12,CGLT14, TT14, Tol14, ADT15, BL14, Bue14, ADT14, AT15, Tol15].The first result of this note is an extension of Pajot’s theorem on rectifiable sets [Paj97]to absolutely continuous rectifiable measures. To state these results, we must recall thenotion of an L p beta number from the theory of quantitative rectifiability. Definition 1.5 ( L p beta numbers) . Let 1 ≤ m ≤ n − ≤ p < ∞ . For everylocally finite Borel measure µ on R n and bounded Borel set Q ⊂ R n , define β ( m ) p ( µ, Q ) by(1.2) β ( m ) p ( µ, Q ) p := inf ℓ Z Q (cid:18) dist( x, ℓ )diam Q (cid:19) p dµ ( x ) µ ( Q ) ∈ [0 , , where ℓ in the infimum ranges over all m -dimensional affine planes in R n . If µ ( Q ) = 0,then we interpret (1.2) as β ( m ) p ( µ, Q ) = 0. WO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES 3
Remark . Beta numbers (of sets) were introduced by Jones [Jon90] to characterizesubsets of rectifiable curves in the plane and are now often called
Jones beta numbers .The L p variant in Definition 1.5 originated in the fundamental work of David and Semmeson uniformly rectifiable sets [DS91, DS93] with the normalization appearing in (1.3). Thenormalization of β ( m ) p ( µ, Q ) presented in Definition 1.5 is not new; see e.g. [Ler03].When Q = B ( x, r ), some sources (e.g. [DS91, DS93, Paj97]) define L p beta numbersusing the alternate normalization(1.3) e β ( m ) p ( µ, B ( x, r )) p := inf ℓ Z B ( x,r ) (cid:18) dist( x, ℓ ) r (cid:19) p dµ ( x ) r m ∈ [0 , ∞ ) , where ℓ in the infimum again ranges over all m -dimensional affine planes in R n . However, β ( m ) p ( µ, B ( x, r )) and e β ( m ) p ( µ, B ( x, r )) are quantitatively equivalent at locations and scaleswhere µ ( B ( x, r )) ∼ r m . We have freely translated beta numbers in theorem statementsquoted from other sources to the convention of Definition 1.5, which is better suited forgeneric locally finite Borel measures. Theorem 1.7 ([Paj97]) . Let ≤ m ≤ n − and let (1.4) (cid:26) ≤ p < ∞ if m = 1 or m = 2 , ≤ p < m/ ( m − if m ≥ .Assume that K ⊂ R n is compact and µ = H m K is a finite measure. If D m ( µ, x ) > at µ -a.e. x ∈ R n and (1.5) Z β ( m ) p ( µ, B ( x, r )) drr < ∞ at µ -a.e. x ∈ R n , then µ is m -rectifiable. In §
2, we note the following extension of Pajot’s theorem. Also, see Theorem 2.1.
Theorem A.
Let ≤ m ≤ n − and let ≤ p < ∞ satisfy (1.4) . Assume that µ is alocally finite Borel measure on R n such that µ ≪ H m . If D m ( µ, x ) > at µ -a.e. x ∈ R n and (1.5) holds, then µ is m -rectifiable. In a forthcoming paper, Tolsa [Tol15] proves that (1.5) is a necessary condition for anabsolutely continuous measure to be rectifiable. Together with Theorem A and (1.1),this result provides a full characterization of absolutely continuous rectifiable measures interms of the beta numbers and lower Hausdorff density of a measure.
Theorem 1.8 ([Tol15]) . Let ≤ m ≤ n − and let ≤ p ≤ . If µ is m -rectifiable and µ ≪ H m , then (1.5) holds. Corollary 1.9.
Let ≤ m ≤ n − and let ≤ p ≤ . If µ is a locally finite Borelmeasure on R n such that µ ≪ H m , then the following are equivalent: • µ is m -rectifiable; • D m ( µ, x ) > at µ -a.e. x ∈ R n and (1.5) holds. In a companion paper to [Tol15], Azzam and Tolsa [AT15] prove that in the case p = 2,Theorem A holds with the hypothesis D m ( µ, x ) > µ -a.e. x ∈ R n on the lower densityreplaced by a weaker assumption D m ( µ, x ) > µ -a.e. x ∈ R n on the upper density. MATTHEW BADGER AND RAANAN SCHUL
For general m -rectifiable measures that are allowed to be singular with respect to H m ,the following basic problem in geometric measure theory is still open. Problem . For all 1 ≤ m ≤ n −
1, find necessary and sufficient conditions in order fora locally finite Borel measure µ on R n to be m -rectifiable. (Do not assume that µ ≪ H m .)Partial progress on Problem 1.10 has recently been made in [GKS10, BS15, AM15] inthe case m = 1. In [GKS10], Garnett, Killip, and Schul exhibit a family ( ν δ ) <δ ≤ δ ofself-similar locally finite Borel measures on R n , which are • doubling : 0 < ν δ ( B ( x, r )) ≤ C δ ν δ ( B ( x, r/ < ∞ for all x ∈ R n and r > • badly linearly approximable : β (1)2 ( ν δ , B ( x, r )) ≥ c δ > x ∈ R n and r > • singular : D ( ν δ , x ) = ∞ at ν δ -a.e. x ∈ R n (hence ν δ ⊥ H ); and, • : ν δ ( R n \ S i Γ i ) = 0 for some countable family of rectifiable curves Γ i .In [BS15], Badger and Schul identify a pointwise necessary condition for an arbitrarylocally finite Borel measure µ on R n to be 1-rectifiable. Theorem 1.11 ([BS15, Theorem A]) . Let n ≥ and let ∆ be a system of closed or half-open dyadic cubes in R n of side length at most 1. If µ is a locally finite Borel measure on R n and µ is 1-rectifiable, then X Q ∈ ∆ β (1)2 ( µ, Q ) diam Qµ ( Q ) χ Q ( x ) < ∞ at µ -a.e. x ∈ R n . The second result of this note is a sufficient condition for a measure µ with D ( µ, x ) = ∞ at µ -a.e. x ∈ R n to be 1-rectifiable. Theorem B.
Let n ≥ and let ∆ be a system of half-open dyadic cubes in R n of sidelength at most 1. If µ is a locally finite Borel measure on R n and (1.6) X Q ∈ ∆ diam Qµ ( Q ) χ Q ( x ) < ∞ at µ -a.e. x ∈ R n , then µ is 1-rectifiable, and moreover, there exist a countable family of rectifiable curves Γ i and Borel sets B i ⊆ Γ i such that H ( B i ) = 0 for all i ≥ and µ ( R n \ S ∞ i =1 B i ) = 0 . Together Theorem 1.11 and Theorem B provide a full characterization of 1-rectifiabilityof measures with “pointwise large beta number” (1.7). Examples of measures that sat-isfy this beta number condition include the measures ( ν δ ) <δ ≤ δ from [GKS10], or moregenerally, any doubling measure µ on R n whose support is R n . Corollary 1.12.
Let n ≥ and let ∆ be a system of half-open dyadic cubes in R n of sidelength at most 1. If µ is a locally finite Borel measure such that (1.7) lim inf Q ∈ ∆ ,x ∈ Q diam Q → β (1)2 ( µ, Q ) > at µ -a.e. x ∈ R n , then µ is 1-rectifiable if and only if (1.6) holds. Finally, we note that in recent work Azzam and Mourgoglou [AM15] give a weakercondition for 1-rectifiability of a doubling measure with connected support.
WO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES 5
Theorem 1.13 ([AM15]) . Let µ be a doubling measure whose support is a topologicallyconnected metric space X and let E ⊆ X be compact. Then µ E is 1-rectifiable if andonly if D ( µ, x ) > for µ -a.e. x ∈ E . When applied to a doubling measure µ on R n whose support is R n , Corollary 1.12 andTheorem 1.13 imply that if D ( µ, x ) > µ -a.e. x ∈ R n , then (1.6) holds.The remainder of this note is split into two sections. We prove Theorem A in § §
3. 2.
Proof of Theorem A
We show how to reduce Theorem A to Theorem 1.7 using standard geometric measuretheory techniques; see Chapters 1, 2, 4, and 6 of [Mat95] for general background. In fact,we will establish the following “localized version” of Theorem A.
Theorem 2.1.
Let ≤ m ≤ n − and let (2.1) (cid:26) ≤ p < ∞ if m = 1 or m = 2 , ≤ p < m/ ( m − if m ≥ .If µ is a locally finite Borel measure on R n such that J p ( µ, x ) := Z β ( m ) p ( µ, B ( x, r )) drr < ∞ at µ -a.e. x ∈ R n , then µ (cid:8) x ∈ R n : 0 < D m ( µ, x ) ≤ D m ( µ, x ) < ∞ (cid:9) is m -rectifiable.Proof. Without loss of generality, we assume for the duration of the proof that H m isnormalized so that ω m = H m ( B m (0 , m . This is the convention used in [Mat95].Suppose that 1 ≤ m ≤ n −
1, let p belong to the range (2.1), and let µ be a locallyfinite Borel measure on R n such that J p ( µ, x ) < ∞ at µ -a.e. x ∈ R n . Define A := (cid:8) x ∈ R n : 0 < D m ( µ, x ) ≤ D m ( µ, x ) < ∞ (cid:9) . Also, for each pair of integers j, k ≥
1, define A ( j, k ) := (cid:8) x ∈ B (0 , k ) : 2 − j r m ≤ µ ( B ( x, r )) ≤ j r m for all 0 < r ≤ − k (cid:9) . Then A ( j, k ) is compact and A ( j, k ) ⊆ A ( j + 1 , k + 1) for all j, k ≥
1. Also note that A = ∞ [ j,k =1 A ( j, k ) = ∞ [ j,k =1 A ( j, k ) , Thus, to prove that µ A is m -rectifiable, it suffices to verify that µ A ( j, k ) is m -rectifiable for all j, k ≥ j, k ≥ K := A ( j, k ), ν := µ K , and σ := H m K . In order toprove that ν is m -rectifiable, it is enough to show that ν ≪ σ ≪ ν and σ is m -rectifiable.By Theorem 6.9 in [Mat95], since 2 − j − − m ≤ D m ( µ, x ) ≤ j +1 − m for all x ∈ K , we have(2.2) ν ( B ( x, r )) = µ ( K ∩ B ( x, r )) ≤ j +1 H m ( K ∩ B ( x, r )) = 2 j +1 σ ( B ( x, r ))and(2.3) σ ( B ( x, r )) = H m ( K ∩ B ( x, r )) ≤ j +1+ m µ ( K ∩ B ( x, r )) = 2 j +1+ m ν ( B ( x, r )) MATTHEW BADGER AND RAANAN SCHUL for all x ∈ R n and r >
0. Note that σ ( R n ) = σ ( B (0 , k )) ≤ j +1+ m µ ( B (0 , k )) < ∞ , since µ is locally finite. That is, σ is a finite measure. Thus, ν and σ are mutuallyabsolutely continuous by (2.2), (2.3), and Lemma 2.13 in [Mat95]. Now,(2.4) σ ( B ( x, r )) ≤ j +1+ m µ ( B ( x, r )) ≤ j +2+ m r m for all x ∈ K and 0 < r ≤ − k − . On the other hand, let K ′ denote the set of x ∈ K such that2 ν ( B ( x, r )) = 2 µ ( K ∩ B ( x, r )) ≥ µ ( B ( x, r )) for all 0 < r ≤ r x for some r x ≤ − k − . Then σ ( R n \ K ′ ) = 0, because ν ( R n \ K ′ ) = µ ( K \ K ′ ) = 0, and(2.5) σ ( B ( x, r )) ≥ − j − µ ( B ( x, r )) ≥ − j − r m for all x ∈ K ′ and 0 < r ≤ r x . In particular, D m ( σ, x ) ≥ c ( m, j ) > σ -a.e. x ∈ R n . To conclude that σ is m -rectifiableusing Theorem 1.7, it remains to verify J p ( σ, x ) < ∞ at σ -a.e. x ∈ R n .By (2.4) and (2.5), there exists a constant C = C ( m, j ) < ∞ such that C − ≤ ν ( B ( x, r )) σ ( B ( x, r )) ≤ C for all 0 < r ≤ r x at σ -a.e. x ∈ R n . Thus, by differentiation of Radon measures, we can write dν = f dσ , where f ∈ L ( dσ )and C − ≤ f ( x ) ≤ C at σ -a.e. x ∈ R n . Therefore, at σ -a.e. x ∈ R n , for every 0 < r ≤ r x and for every m -dimensional affine plane ℓ , Z B ( x,r ) (cid:18) dist( y, ℓ )diam B ( x, r ) (cid:19) p dσ ( y ) σ ( B ( x, r )) ≤ C Z B ( x,r ) (cid:18) dist( y, ℓ )diam B ( x, r ) (cid:19) p dν ( y ) ν ( B ( x, r )) ≤ C Z B ( x,r ) (cid:18) dist( y, ℓ )diam B ( x, r ) (cid:19) p dµ ( y ) µ ( B ( x, r )) . Thus, β ( m ) p ( σ, B ( x, r )) ≤ (2 C ) /p β ( m ) p ( µ, B ( x, r )) for all 0 < r ≤ r x at σ -a.e. x ∈ R n .Since J p ( µ, x ) < ∞ at µ -a.e. x ∈ R n and σ ≪ µ , it follows that J p ( σ, x ) < ∞ at σ -a.e. x ∈ R n . Finally, since K is compact, σ = H m K is finite, and D m ( σ, x ) > J p ( σ, x ) < ∞ at σ -a.e. x ∈ R n , we conclude that σ is m -rectifiable by Theorem 1.7.As noted above, this implies that ν = µ A ( j, k ) is m -rectifiable for all j, k ≥
1, andtherefore, µ A is m -rectifiable. (cid:3) Proof of Theorem B
For every Borel measure µ on R n , define the quantity S ( µ, x ) := X Q ∈ ∆ diam Qµ ( Q ) χ Q ( x ) ∈ [0 , ∞ ] for all x ∈ R n , where ∆ denotes any system of half-open dyadic cubes in R n of side length at most 1.Theorem B is a special case of the following statement. Theorem 3.1.
Let n ≥ . If µ is a locally finite Borel measure on R n , then ρ := µ { x ∈ R n : S ( µ, x ) < ∞} is -rectifiable. Moreover, there exists a countable family of rectifiable curves Γ i ⊂ R n andBorel sets B i ⊆ Γ i such that H ( B i ) = 0 for all i ≥ and ρ ( R n \ S ∞ i =1 B i ) = 0 . WO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES 7
We start with a lemma, which will be used to organize the proof of Theorem 3.1.
Lemma 3.2.
Let n ≥ and let µ be a locally finite Borel measure on R n . Given Q ∈ ∆ such that η := µ ( Q ) > and N < ∞ , let A := { x ∈ Q : S ( µ, x ) ≤ N } . For all < ε < /η , the set of dyadic cubes Q ⊆ Q can be partitioned into good cubes and bad cubes with the following properties: (1) every child of a bad cube is a bad cube; (2) the set B := A \ S { Q : Q ⊆ Q is a bad cube } satisfies µ ( B ) ≥ (1 − εη ) µ ( A ) ; (3) P diam Q < N/ε , where the sum ranges over all good cubes Q ⊆ Q .Proof. Suppose that n , µ , Q , η , N , and A are given as above and let ε >
0. If µ ( A ) = 0,then we may declare every dyadic cube Q ⊆ Q to be a bad cube and the conclusionof the lemma hold trivially. Thus, suppose that µ ( A ) >
0. Declare that a dyadic cube Q ⊆ Q is a bad cube if there exists a dyadic cube R ⊆ Q such that Q ⊆ R and µ ( A ∩ R ) ≤ εµ ( A ) µ ( R ). We call a dyadic cube Q ⊆ Q a good cube if Q is not a badcube. Property (1) is immediate. To check property (2), observe that µ ( A \ B ) ≤ X maximal bad Q ⊆ Q µ ( A ∩ Q ) ≤ εµ ( A ) X maximal bad Q ⊆ Q µ ( Q ) ≤ εµ ( A ) µ ( Q ) , where the last inequality follows because the maximal bad cubes are pairwise disjoint(since ∆ is composed of half-open cubes). Recalling µ ( Q ) = η , it follows that µ ( B ) = µ ( A ) − µ ( A \ B ) ≥ (1 − εη ) µ ( A ) . Thus, property (2) holds. Finally, since S ( µ, x ) ≤ N for all x ∈ A , N µ ( A ) ≥ Z A S ( µ, x ) dµ ( x ) ≥ X Q ⊆ Q diam Q µ ( A ∩ Q ) µ ( Q ) > εµ ( A ) X good Q ⊆ Q diam Q, where we interpret µ ( A ∩ Q ) /µ ( Q ) = 0 if µ ( Q ) = 0. Because µ ( A ) >
0, it follows that X good Q ⊆ Q diam Q < Nε .
This verifies property (3). (cid:3)
Lemma 3.3.
Let n ≥ and let µ be a locally finite Borel measure on R n . If µ ( { x ∈ Q : S ( µ, x ) ≤ N } ) > for some Q ∈ ∆ and N < ∞ , then for all < ε < /µ ( Q ) the set B = B ( µ, Q , N, ε ) described in Lemma 3.2 lies in arectifiable curve Γ with H (Γ) < N/ ε and H ( B ) = 0 .Proof. Let n ≥ µ be a locally finite Borel measure on R n . Suppose µ ( A ) > Q ∈ ∆ and N < ∞ , where A = { x ∈ Q : S ( µ, x ) ≤ N } . Then η := µ ( Q ) > < ε < /η , let B = B ( µ, Q , N, ε ) denote the set from Lemma 3.2.Since ε is small enough such that µ ( B ) ≥ (1 − εη ) µ ( A ) >
0, the cube Q is a good cube.Construct a connected set T ⊂ R n by drawing a (closed) straight line segment ℓ Q from MATTHEW BADGER AND RAANAN SCHUL the center of each good cube Q ( Q to the center of its parent, which is also a goodcube. Let T denote the closure of T . For all δ > T ⊆ [ good Q ( Q diam Q>δ ℓ Q ∪ [ good Q ⊆ Q diam Q ≤ δ Q, whence H δ ( T ) ≤ X good Q ( Q diam Q>δ diam ℓ Q + X good Q ⊆ Q diam Q ≤ δ diam Q = X good Q ( Q diam Q>δ
12 diam Q + X good Q ⊆ Q diam Q ≤ δ diam Q. Here we used the fact that any straight line segment ℓ can be subdivided into finitelymany line segments ℓ ′ , . . . , ℓ ′ k such that diam ℓ ′ i ≤ δ for all i and P ki =1 diam ℓ ′ i = diam ℓ .Since P good Q ⊆ Q diam Q < N/ε , it follows that H ( T ) = lim δ ↓ H δ ( T ) ≤ X good Q ( Q diam Q < N ε . Now, B ⊆ Q \ [ bad Q ⊂ Q Q = [ ( ∞ \ i =0 Q i : Q ⊇ Q ⊇ · · · is a chain of good cubes, lim i →∞ diam Q i = 0 ) (3.1) ⊆ n lim i →∞ x i : x i ∈ ℓ Q i for some good cubes Q ⊇ Q ⊇ · · · , lim i →∞ diam Q i = 0 o . (3.2)Thus, B ⊆ T by (3.2). Moreover, refining (3.1), we obtain B ⊆ T ∞ j =1 G j , where G j = [ ( ∞ \ i = j Q ′ i : Q ′ j ) Q ′ j +1 ) · · · is a chain of good cubes, diam Q ′ j ≤ − j ) . Since P good Q ⊆ Q diam Q < ∞ , we have H − j ( G j ) →
0, which implies H ( B ) = 0. Finally,because T is a continuum in R n with H ( T ) < ∞ , T coincides with the image Γ = f ([0 , f : [0 , → R n ; e.g. see [DS93, Theorem I.1.8] or [Sch07, Lemma3.7]. (cid:3) The proof of Theorem 3.1 uses Lemmas 3.2 and 3.3 repeatedly over a suitable, countablechoice of parameters.
Proof of Theorem 3.1.
Suppose n ≥ µ be a locally finite Borel measure on R n .Our goal is to show that µ { x ∈ R n : S ( µ, x ) < ∞} is 1-rectifiable. It suffices to provethat µ { x ∈ Q : S ( µ, x ) ≤ N } is 1-rectifiable for all Q ∈ ∆ and for all integers N ≥ Q ∈ ∆ and N ≥
1. Let A = { x ∈ Q : S ( µ, x ) ≤ N } . If µ ( A ) = 0, then thereis nothing to prove. Thus, assume µ ( A ) >
0. Then η = µ ( Q ) >
0, as well. Pick anysequence ( ε i ) ∞ i =1 such that 0 < ε i < /η for all i ≥ ε i → i → ∞ . By Lemmas 3.2 WO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES 9 and 3.3, there exist a Borel set B i = B ( µ, Q , N, ε i ) ⊆ A and a rectifiable curve Γ i ⊇ B i such that H ( B i ) = 0 and µ ( A \ B i ) ≤ ε i ηµ ( A ). Hence µ A \ ∞ [ i =1 Γ i ! ≤ µ A \ ∞ [ i =1 B i ! ≤ inf j ≥ µ ( A \ B j ) ≤ ηµ ( A ) inf j ≥ ε j = 0 . Therefore, µ A is 1-rectifiable, and moreover, µ A ( R n \ S ∞ i =1 B i ) = 0. (cid:3) References [ADT14] Jonas Azzam, Guy David, and Tatiana Toro,
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Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009
E-mail address : [email protected] Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651
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