Rectifiability of sets and measures - University of …toro/Courses/15-16/...References P. Mattila....

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Rectifiability of sets and measures

Tatiana Toro

University of Washington

February 15, 2016

Tatiana Toro (University of Washington) Historic recollection February 15, 2016 1 / 1

References

P. Mattila. Geometry of sets and measures in Euclidean spaces,Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge,1995.

D. Preiss, Geometry of measures in Rn: distribution, rectifiability, anddensities, Ann. of Math. 125 (1987), 537-643.

D. Preiss, X. Tolsa and T. Toro, On the smoothness of Holderdoubling measures, Calculus of Variations and PDE’s 35 (2009),339-363.

Geometry of Measures

In the 1920’s Besicovitch studied sets with locally finite “length” in theplane.What is the length of a set?

The basic question: do the infinitesimal properties of the “length” of a setE in the plane yield geometric information about E itself?What are the infinitesimal properties of the ‘length” of a set E? What sortof geometric information can one expect?

This simple question marks the beginning of the study of the geometry ofmeasures and the associated field known as Geometric Measure Theory!

Hs : s-dimensional Hausdorff measure in Rm

Let A ⊂ Rm, 0 ≤ s <∞, 0 < δ ≤ ∞

Hsδ(A) := inf

∞∑j=1

(diamCj)s |A ⊂

∞⋃j=1

Cj , diamCj ≤ δ

Hs(A) = lim

δ→0Hsδ(A) = sup

δ>0Hsδ(A).

Properties:

Hs is a Borel measure.

Hm = Lm (m-Lebesgue measure) in Rm.

Hs ≡ 0 for s > m.

Hs(λA) = λsHs(A) for λ > 0.

dimA := inf {0 ≤ s <∞|Hs(A) = 0}

Examples

Hausdorff dimension 1 Hausdorff dimensionlog 4log 3

Structure Theorem

Besicovitch (1929-1939) Suppose X ⊂ R2 and 0 < H1(X ) <∞. Then

X = Y ∪ Z

Y ⊂⋃∞

i=1 Γi , where Γi is a Lipschitz image of R.

H1(πL(Z )) = 0 for almost every line L ⊂ R2. Here πL denotes theorthogonal projection of R2 onto L.

Federer (1947) Suppose X ⊂ Rm, n ∈ N and 0 < Hn(X ) <∞. Then

X = Y ∪ Z

Y ⊂⋃∞

i=1 Σi , where Σi is a Lipschitz image of Rn

Hn(πL(Z )) = 0 for almost every n-dimensional plane L ⊂ Rm. HereπL denotes the orthogonal projection of Rm onto L.

Rectifiable sets

A map f : Rn → Rm is Lipschitz if there exists M > 0 such that forx , y ∈ Rn

|f (x)− f (y)| ≤ M|x − y |

Σ = f (Rn) is a Lipschitz image of Rn.

E ⊂ Rm is n-rectifiable, n ∈ {1, · · · ,m} if there exists a family {Σi}iof Lipschitz images of Rn such that

∞⋃i=1

( ∞⋃i=1

)∪ Σ0

with Hn(Σ0) = 0.

Remarks

By Whitney’s extension theorem E ⊂ Rm is n-rectifiable iff thereexists a family {Σi}i n-dimensional smooth sub-manifolds and a setΣ0 such that

( ∞⋃i=1

)∪ Σ0

with Hn(Σ0) = 0.

In 1998 B. White showed using a “slicing” argument that Federer’sresult can be deduced from Besicovitch.

Purely unrectifiable sets

Z ⊂ Rm is n-purely unrectifiable if 0 < Hn(Z ) <∞ andHn(πL(Z )) = 0 for almost every n-dimensional plane L ⊂ Rm.

Example: 4-corner Cantor set

I There exists C > 1 such that for each x ∈ E∞ and r ∈ (0,√

C−1r ≤ H1(E∞ ∩ B(x , r)) ≤ Cr

I For almost every line L in R2, H1(πL(E∞)) = 0.

Structure Theorem

Besicovitch (1929-1939) Suppose X ⊂ R2 and 0 < H1(X ) <∞. Then

X = Y ∪ Z

Y is 1-rectifiable.

Z is 1-purely unrectifiable.

Federer (1947) Suppose X ⊂ Rm, n ∈ N and 0 < Hn(X ) <∞. Then

X = Y ∪ Z

Y is n-rectifiable.

Z is n-purely unrectifiable.

Rectifiable measures

A measure µ in Rm is Radon if it is Borel regular and µ(K ) <∞ foreach compact subset K ⊂ Rm.

Federer A Radon measure µ in Rm is n-rectifiable iff there exists afamily {Σi}i of Lipschitz images of Rn such that

µ(Rm\∞⋃i=1

Σi ) = 0

Mattila & Preiss A Radon measure µ in Rm is n-rectifiable iffµ� Hn and there exists a family {Σi}i of Lipschitz images of Rn

such that

µ(Rm\∞⋃i=1

Σi ) = 0

Remarks

If E ⊂ Rm Borel with 0 < Hn(E ) <∞, Hn E defined byHn E (A) = Hn(E ∩ A) is a Radon measure. The rectifiablity of Eis equivalent to the rectifiability of Hn E (in the sense of Mattila &Preiss)

How different are these two notions of rectifiability?

I Garnett-Killip-Schul: There exists a 1-rectifiable doubling measure µ(in the sense of Federer) supported in all of Rm. Hence H1 ⊥ µ.

Garnett-Killip-Schul example in R2

First iterations

Courtesy of M. Badger

Properties of the Garnett-Killip-Schul example

sptµ = {x ∈ Rm : µ(B(x , s)) > 0, ∀s > 0} = Rm

µ is doubling, i.e. there exists C > 0 such that for x ∈ sptµ = Rm

and s > 0µ(B(x , 2s)) ≤ Cµ(B(x , s))

If f : R→ Rm and Γ = {(x , f (x)) : x ∈ R} then µ(Γ) = 0.

There exists a family {Σi}i of Lipschitz images of R such that

µ(Rm\∞⋃i=1

Σi ) = 0.

What is the difference between a Lipschitz graph and a Lipschitzimage?

Besicovitch question

Let E ⊂ R2 such that 0 < H1(E ) <∞. Suppose that

limr→0

H1(E ∩ B(x , r))

2r= 1 H1 − a.e. x ∈ E . (1)

What can be said about the structure of E?

Examples of sets satisfying (1):

E ⊂ R× {0} Borel, 0 < H1(E ) <∞

E C 1 curve.

Examples of sets not satisfying (1):

4-corner Cantor set E∞.

Densities

For a Radon measure µ in Rm we say that the n-density at x exists if

limr→0

µ(B(x , r))

rn= θn(µ, x) ∈ (0,∞)

What can be said about a Radon measure µ for which the n-densityexists µ-a.e. x ∈ Rm?

Radon measures with n-density in Rm are n-rectifiable

µ = Hn E µ

1928-1938 Besicovitchn = 1, m = 2

1944 Morse & Randolphn = 1, m = 2

E n-rectifiable µ n-rectifiable

1947 Federer: If Hn E Radon andE n-rectifiable then

θn(Hn E , x) = ωn, Hn − a.e. x ∈ E

ωn = measure of the unit ball in Rn

Radon measures with n-density in Rm are n-rectifiable

µ = Hn E µ

1928-1938 Besicovitchn = 1, m = 2

1944 Morse & Randolphn = 1, m = 2

1950 Mooren = 1, m ≥ 3

1961 Marstrandn = 2, m = 3

1975 Mattilan ≤ m, n,m ∈ N

1987 Preissn,m ∈ N

E n-rectifiable µ n-rectifiable

Theorem (Besicovitch) Let E ⊂ R2 such that 0 < H1(E ) <∞ and

limr→0

H1(E ∩ B(x , r))

2r= 1 H1 − a.e. x ∈ E ,

then E is 1-rectifiable.

Theorem (Preiss) Let µ be a Radon measure in Rm such that

limr→0

µ(B(x , r))

rn= θn(µ, x) ∈ (0,∞)

for µ-a.e. x ∈ Rm. Then µ is n-rectifiable.

Questions

1 Are there conditions that ensure the Federer-rectifiability of ameasure? (Badger-Schul n = 1).

2 Are there conditions that ensure the Mattila-Preiss-rectifiability of ameasure?

3 What happens when we consider the s-density for s 6∈ N?

4 Does the density ratio µ(B(x ,r))rn encode further regularity for µ?

Questions about infinitesimal structure of a measure.How do we study infinitesimal structure of mathematical objects?

Weak convergence of Radon measures

Let µ, µi (i = 1, 2, · · · ) be Radon measures in Rm. The following areequivalent

limi→∞

ˆϕ dµi =

ˆϕ dµ for all ϕ ∈ Cc(Rm)

lim supi→∞

µi (K ) ≤ µ(K ) for each compact set K ⊂ Rm and

lim infi→∞

µi (U) ≤ µ(U) for each open set U ⊂ Rm.

limi→∞

µi (B) = µ(B) for each Borel set B ⊂ Rm with µ(∂B) = 0.

Notation: µi ⇀ µ.

Compactness of Radon measures

Let {µi}i be a sequence of Radon measures in Rm such that for eachcompact set K ⊂⊂ Rm

supiµi (K ) <∞.

There exists a subsequence {µik}ik and a Radon measure µ such that

µik ⇀ µ.

Remark: The set of Radon measures in Rm can be metrized in sucha way that it becomes a complete metric space and the metricconvergence coincides with the weak convergence.

Tools to study infinitesimal structures: dilations

The map Ta,r (x) = x−ar maps B(a, r) onto B(0, 1).

The image of a Radon measure µ under Ta,r is the Radon measureTa,r [µ] defined by

Ta,r [µ](E ) = µ(rE + a)

where rE + a = {rx + a : x ∈ E}. Note

Ta,r [µ](B(0, 1)) = µ(B(a, r)).

Tangent measures

A non-zero Radon measure ν is a tangent measure to µ at a ∈ Rm if thereexist sequences {ri}i , ri → 0 and {ci}i , ci > 0 such that

ciTa,ri [µ] ⇀ ν as i →∞,

i.e. ∀ϕ ∈ Cc(Rm)

limi→∞

x − a

ri) dµ(x) =

ˆϕ dν.

The set of all tangent measures to µ at a is denoted by

Tan(µ, a).

Examples

If A ⊂ Rm Borel measurable, µ = Hm A by Lebesgue densitytheorem for µ a.e. a ∈ Rm

Tan(µ, a) = {cHm : c > 0}.

If f ∈ L1loc(Rm), f > 0 and µ(E ) =´E f (x) dx then for µ a.e. a ∈ Rm

Tan(µ, a) = {cHm : c > 0}.

For ϕ ∈ Cc(Rm)

x − a

ri)f (x) dx = ci

ˆϕ(y)f (riy + a)rmi dy

= ci rmi

ˆϕ(y)(f (riy + a)− f (a)) dy

+ci rmi f (a)

ˆϕ(y) dy

· · ·

Examples

If A ⊂ Rm Borel measurable, µ = Hm A by Lebesgue densitytheorem for µ a.e. a ∈ Rm

Tan(µ, a) = {cHm : c > 0}.

If f ∈ L1loc(Rm), f > 0 and µ(E ) =´E f (x) dx then for µ a.e. a ∈ Rm

Tan(µ, a) = {cHm : c > 0}.

If Σ ⊂ Rm a C 1 n-dimensional submanifold and µ = Hm Σ then

Tan(µ, a) = {cHn (TaΣ− a) : c > 0}

where TaΣ is the tangent plane to Σ at a.

Mattila constructed a family of examples where the tangents are notunique and none of them are flat either.

Existence

Preiss (1987) For any Radon measure µ, Tan(µ, a) 6= ∅, µ- a.e.a ∈ Rm.

Let µ be a Radon measure in Rm, if for a ∈ Rm

ca = lim supr→0

µ(B(a, 2r))

µ(B(a, r))<∞ (µ asymptotically doubling at a)

then every sequence {ri}i , ri → 0 contains a subsequence {rik}ik suchthat

µa,rik =1

µ(B(a, rik ))Ta,rik

[µ] ⇀ ν 6= 0 and ν ∈ Tan(µ, a).

Moreover any tangent measure is obtained this way up to a constant.

Rectifiability of sets and measures - University of …toro/Courses/15-16/...References P. Mattila....

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