Structure theorems for Radon measures Matthew Badger Department of Mathematics University of Connecticut Analysis on Metric Spaces University of Pittsburgh March 10 11, 2017 Research Partially Supported by NSF DMS 1500382, 1650546
Decomposition of Measures Geometric Measure Theory: Understand a measure on a space through its interaction with canonical sets in the space Let (, M, µ) be a measure space Let N M be a family of measurable sets µ is carried by N if there exist countably many sets Γ N such that µ ( \ Γ ) = 0 µ is singular to N if µ(γ) = 0 for every Γ N Exercise (Decomposition Lemma) If µ is σ-finite, then µ can be written uniquely as µ N + µ N where µ N is carried by N and µ N is singular to N Proof of the Decomposition Theorem is abstract nonsense Identification Problem: Find measure-theoretic and/or geometric characterizations or constructions of µ N and µ N?
PSA: Don t Think About Support Three Measures Let > 0 be weights with =1 = 1 Let { : 1}, {l : 1}, { : 1} be a dense set of points, unit line segments, unit squares in the plane µ 0 = =1 δ µ 1 = =1 1 l µ 2 = =1 2 µ 0, µ 1, µ 2 are probability measures on R 2 µ is smallest closed set carrying µ; µ 0 = µ 1 = µ 2 = R 2 µ is carried by -dimensional sets (points, lines, squares) The support of a measure is a rough approximation that hides the underlying structure of a measure
Example: Atomic Measures vs Atomless Measures A Radon measure is a locally finite, Borel regular outer measure Let µ be a Radon measure on R Then we can write µ = µ 0 + µ 0, where µ 0 is carried by singletons (ie µ 0 is atomic) µ 0 is singular to singletons (ie µ 0 is atomless) How can you identify µ 0 and µ 0? µ 0 = µ { R : 0 µ( (, )) > 0} µ 0 = µ { R : 0 µ( (, )) = 0} µ denotes the restriction of µ to : (µ )( ) = µ( )
1-Rectifiable and Purely 1-Unrectifiable Measures A singleton { } has the following properties: { } is connected, compact, H 0 ({ }) < A candidate Γ for a 1-dimensional atom might satisfy: Γ is connected, compact, H 1 (Γ) < Theorem A set Γ R is connected, compact, and H 1 (Γ) < if and only if there exists a Lipschitz map : [0, 1] R such that Γ = ([0, 1]) A connected, compact set Γ R with H 1 (Γ) < is called a rectifiable curve or Lipschitz curve Decomposition: Let µ be a Radon measure on R Then µ = µ 1 + µ 1, where µ 1 is carried by rectifiable curves (µ 1 is 1-rectifiable) µ 1 is singular to rectifiable curves (µ 1 is purely 1-unrectifiable)
Identification Problem for µ 1 and µ 1 If R and H 1 ( ) <, then 1 H 1 ( (, )) 1 2 0 2 H 1 -ae Solution for Hausdorff measures: Theorem (Besicovitch 1928,1938) Let µ = H 1, where R 2 and 0 < H 1 ( ) < Then µ 1 = µ { } R 2 µ( (, )) : = 1 0 2 = µ { R 2 : has a tangent line at } µ 1 = µ = µ { R 2 µ( (, )) : 3 } 0 2 4 { R 2 : does not have a tangent line at }
Identification Problem for µ 1 and µ 1 Let µ be a Radon measure on R The following are equivalent: µ H 1 (ie H 1 ( ) = 0 µ( ) = 0) µ( (, )) 0 < at µ-ae R 2 Solution for absolutely continuous measures: Theorem (Morse and Randolph 1944) Let µ be a Radon measure on R 2 with µ H 1 Then µ 1 = µ { } R 2 µ( (, )) : 0 < < 0 2 = µ { R 2 : has a µ-approximate tangent line at } µ 1 = µ = µ { } R 2 µ( (, )) : 1 µ( (, )) 0 2 101 0 2 { R 2 : does not have a µ-approximate tangent line at }
Examples I For 1, let be a subset of a rectifiable curve Γ R µ = =1 H1 is a finite 1-rectifiable measure, µ H 1 Let be the Cantor middle-third set and let µ = H {0}, where = (2)/ (3) is the Hausdorff dimension of µ is 1-rectifiable measure with µ( (, )) for all µ and 0 < 1; and µ H 1 µ( (, )) and 0 2 = µ-ae
Examples II Lebesgue measure on R is purely 1-unrectifiable for all 2 (This is obvious!) Let R 2 be the 4 corners Cantor set, = =0 is Ahlfors regular: H 1 ( (, )) for all, 0 < 1 Every rectifiable curve Γ = ([0, 1]) R 2 intersects in a set of zero H 1 measure H 1 is a purely 1-unrectifiable measure on R 2
Examples III Let λ R 2 be the generalized 4 corners Cantor set, where 0 < λ 1/2 is the scaling factor (λ = 1/5) has Hausdorff dimension = (4)/ (1/λ) H ( λ (, )) for all λ and 0 < 1 When λ = 1/2, = 2 and H is just Lebesgue measure restricted to the unit square If 1/4 λ 1/2, then H λ is purely 1-unrectifiable If 0 < λ < 1/4, then H λ is 1-rectifiable see eg Martin and Mattila (1988)
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable
Examples IV Theorem (Garnett-Killip-Schul 2010) There exist a Radon measure µ on R 2 with µ = R 2 such that µ is doubling (µ( (, 2 )) µ( (, ))), µ H 1, and µ is 1-rectifiable µ( (, )) 0 = µ-ae 2 1 ( ) 1 µ( (, )) 2 < µ-ae 0 (see B-Schul 2016) µ(γ) = 0 whenever Γ = ([0, 1]) and : [0, 1] R 2 is bi-lipschitz Nevertheless there exist Lipschitz maps ( : [0, 1] R 2 such ) that µ R 2 \ ([0, 1]) = 0 =1
Identification Problem for µ 1 and µ 1 A Radon measure µ on R is doubling if µ( (, 2 )) µ( (, )) for all µ and > 0 Solution for doubling measures with connected supports: Theorem (Azzam-Mourgoglou 2016) Let µ be a doubling measure on R such that µ is connected Then { } µ 1 = µ R µ( (, )) : > 0 0 2 µ 1 = µ { } R µ( (, )) : = 0 0 2 Azzam and Mourgoglou s main result applies to doubling measures with connected supports on arbitrary metric spaces
Non-homogeneous 2 Jones β numbers Let µ be a Radon measure on R For every cube, define β 2 (µ, 3 ) = line β 2 (µ, 3, ) [0, 1], where β 2 (µ, 3, ) 2 = 3 ( (, ) 3 ) 2 µ( ) µ(3 ) Non-homogeneous refers to the normalization 1/µ(3 )
Identification Problem for µ 1 and µ 1 A Radon measure µ on R is pointwise doubling if 0 µ( (, 2 )) µ( (, )) < at µ-ae R Solution for pointwise doubling measures: Theorem (B and Schul 2017) Let µ be a pointwise doubling measure on R Then { µ 1 = µ R : } 2 (µ, ) < { µ 1 = µ R : } 2 (µ, ) = Here 2 (µ, ) is a non-homogeneous 2 Jones square function: 2 (µ, ) = dyadic 1 β 2 (µ, 3 ) 2 µ( ) χ ( )
Interpretation Let µ be a pointwise doubling measure on R Then µ is 1-rectifiable iff 2(µ, ) = dyadic 1 β 2(µ, 3 ) 2 µ( ) χ ( ) < µ-ae Extreme Behaviors Suppose 0 < 0 µ( (, )) 2 0 µ( (, )) 2 Then µ is 1-rectifiable if and only if β 2(µ, 3 ) 2 χ ( ) < µ-ae < µ-ae dyadic 1 When µ = H 1, R compact, this was proved by Pajot (1997) Suppose µ has is badly linearly approximable in the sense β 2(µ, 3 ) > 0 µ-ae Then µ is 1-rectifiable if and only if dyadic 1 µ( ) χ ( ) < µ-ae
Examples V Theorem (Martikainen and Orponen, arxiv 2016) For all ε > 0, there exists a probability measure µ on R 2 with µ [0, 1] 2 such that 1 pointwise non-doubling: 0 µ( (, 2 )) µ( (, )) = µ-ae µ( (, )) 2 vanishing lower density: 0 = 0 µ-ae 2 3 uniformly bounded square function: 2 (µ, ) = dyadic 1 2 β 2 (µ, 3 ) µ( ) χ ( ) ε for all µ Interpretation: vanishing lower density implies that µ is purely 1-unrectifiable cannot hope to characterize rectifiability of a Radon measure using only non-homogeneous square function 2 (µ, )
Anisotropic 2 Jones β numbers Given dyadic cube in R, ( ) denotes a subdivision of 1600 into overlapping dyadic cubes of same / previous (larger) generation as For every Radon measure µ on R and every dyadic cube, we define β 2 (µ, ) 2 = line ( ) β 2(µ, 3, ) 2 3 [0, 1], where β 2(µ, 3, ) 2 3 = 3 ( ) 2 ( ) (, ) µ(3 ) µ( ) 1, 3 3 µ(3 )
Identification Problem for µ 1 and µ 1 Solution for Radon measures: Theorem (B and Schul 2017) Let µ be a Radon measure on R Then { } µ 1 = µ R µ( (, )) : > 0 and 2 (µ, ) < 0 2 µ 1 = µ { } R µ( (, )) : = 0 or 2 (µ, ) = 0 2 Here 2 (µ, ) is an anisotropic 2 Jones square function: 2 (µ, ) = dyadic 1 β 2(µ, ) 2 µ( ) χ ( )
Main Takeaway Corollary A Radon measure µ on R is 1-rectifiable if and only if at µ-ae, µ( (, )) 0 > 0 and 2 2 (µ, ) = dyadic 1 β 2 (µ, ) 2 µ( ) χ ( ) < A Radon measure µ on R is purely 1-rectifiable if and only if at µ-ae, µ( (, )) 0 = 0 or 2 2 (µ, ) = dyadic 1 β 2 (µ, ) 2 µ( ) χ ( ) = Takeaway: To understand geometric properties of non-doubling measures (such as rectifiability) using multiscale analysis, it may be convenient or necessary to incorporate anisotropic normalizations
Proof Ingredient: Drawing Rectifiable Curves Theorem (B and Schul 2017) Let 2, let > 1, let 0 R, and let 0 > 0 Let ( ) =0 be a sequence of nonempty finite subsets of ( 0, 0 ) such that 1 distinct points, are uniformly separated: 2 0 ; 2 for all, there exists +1 +1 such that +1 < 2 0 ; and, 3 for all ( 1), there exists 1 1 such that 1 < 2 0 Suppose that for all 1 and for all we are given a straight line l, in R and a number α, 0 such that (, l, ) α, 2 0 (1) ( 1 ) (,65 2 0 ) and α 2, 2 0 < (2) =1 Then the sets converge in the Hausdorff metric to a compact set ( 0, 0 ) and there exists a compact, connected set Γ ( 0, 0 ) such that Γ and H 1 (Γ), 0 + α 2, 2 0 (3) =1 This is a flexible criterion for drawing a rectifiable curve through the leaves of a tree; extends P Jones Traveling Salesman construction (Inventiones 1990), which required +1 Our write-up separates relatively simple description of the curve from the intricate length estimates
Sample of Higher Dimensional Results I A Radon measure µ on R is -rectifiable if µ(r \ Γ ) = 0 for some sequence of images Γ of Lipschitz maps : [0, 1] R That is, a Radon measure is -rectifiable provided it is carried by Lipschitz images of -cubes Theorem (Preiss 1987) Assume 0 µ( (, )) < µ-ae (or equivalently, µ H ) Then µ is -rectifiable if and only if 0 < 0 µ( (, )) < Preiss introduced tangent measures and studied global geometry of -uniform measures in R New examples of 3-uniform measures in R have been announced by Nimer on the arxiv in August 2016!!!!!
Sample of Higher Dimensional Results II A Radon measure µ on R is -rectifiable if µ(r \ Γ ) = 0 for some sequence of images Γ of Lipschitz maps : [0, 1] R Theorem (Azzam-Tolsa + Tolsa 2015) µ( (, )) Assume 0 < 0 < µ-ae Then µ is -rectifiable if and only if the homogeneous 2 Jones function 2 (µ, ) = 1 0 β ( ) 2 (µ, (, )) 2 µ( (, )) < µ-ae One ingredient in Azzam-Tolsa s proof is David and Toro s version of the Reifenberg algorithm for sets with holes Edelen-Naber-Valtorta announced an extension of Azzam-Tolsa on the arxiv in December 2016
Understanding higher-dim rectifiability is hard A Radon measure µ on R is -rectifiable if µ(r \ Γ ) = 0 for some sequence of images Γ of Lipschitz maps : [0, 1] R This definition is due to Federer (1947) When 2, it is not clear if Lipschitz images of [0, 1] is the correct family N of -dimensional sets The catastrophe: If : [0, 1] R is Lipschitz, then Γ = ([0, 1] ) is connected, compact, and H (Γ) < But the converse is false when 2! Open Problem: Find additional metric, geometric, and/or topological conditions which ensure that a compact, connected set R with H 2 ( ) < is contained in the image of a Lipschitz map : [0, 1] 2 R
Related developments Azzam and Schul, The Analyst s Traveling Salesman Theorem for sets of dimension larger than one, arxiv 2016 Novel definition of higher-dimensional beta numbers of sets using the Choquet integral Results include a characterization of subsets of Reifenberg vanishing bi-lipschitz surfaces that is similar to Jones TST K Rajala, Uniformization of two-dimensional metric surfaces, Inventiones 2016 Gives an intrinsic characterization of metric spaces with locally finite H 2 measure that are quasiconformally equivalent to R 2 Does not immediately extend to higher dimensions
Current Project (w/ Vellis): Non-integral Dimensions For each [1, ], let N denote all (1/ )-Hölder curves in R, ie all images Γ of (1/ )-Hölder continuous maps : [0, 1] R Decomposition: Every Radon measure µ on R can be uniquely written as µ = µ N + µ N, where µ N is carried by (1/ )-Hölder curves µ N Notes is singular to (1/ )-Hölder curves Every measure µ on R is carried by (1/ )-Hölder curves (space-filling curves) A measure µ is carried by 1-Hölder curves iff µ is 1-rectifiable If µ is -rectifiable, then µ is carried by (1/ )-Hölder curves Martín and Mattila (1988,1993,2000) studied this concept for measures µ of the form µ = H, where 0 < H ( ) <
Measures with extreme lower densities Theorem (B-Vellis, in preparation) Let µ be a Radon measure on R and let (1, ) Then the measure } µ { := µ R µ( (, )) : = 0 0 0 is singular to (1/ )-Hölder curves, ie µ (Γ) = 0 whenever Γ is a (1/ )-Hölder 0 curve; and, the measure 1 } µ { := µ R µ( (, 2 )) : < and µ( (, )) 0 µ( (, )) < 0 is carried by H null sets of (1/ )-Hölder curves, ie there exist (1/ )-Hölder curves and Borel sets Γ with H ( ) = 0 such that µ (R \ =1 ) = 0 1 The condition 0 µ( (, )) < implies µ( (, )) 0 = The theorem is also true when = 1 by B-Schul (2015, 2016)
Further results Corollary (B-Vellis, in preparation) Let µ be a Radon measure on R, let [1, ), and let [0, ) Then the measure µ + := µ { } R µ( (, )) µ( (, )) : 0 < < 0 0 is carried by H null sets of (1/ )-Hölder curves Theorem (B-Vellis, in preparation) Let µ be a Radon measure on R and let [0, 1) Then the measure µ + is carried by H 1 null sets of bi-lipschitz curves, ie there exist bi-lipschitz curves Γ and Borel sets Γ with H 1 ( ) = 0 st µ +(R \ =1 ) = 0 Martìn and Mattila (1988): If 0 < H ( ) < for some [0, 1) and 0 H ( (, )) > 0 at H -ae, then H is 1-rectifiable
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