An introduction to Geometric Measure Theory Part 1 ...mcs.open.ac.uk/tcon2/teaching/LTCC-2016-week-1.pdfAn introduction to Geometric Measure Theory Part 1: dimension 3 October 2016

An introduction to Geometric Measure TheoryPart 1: dimension3 October 2016

Toby O’Neil, October 2016

TCON (Open University) An introduction to GMT, part 1 October 2016 1 / 56

Proposed planWeek 1 Introduce notions of dimensionWeek 2 Develop theory of dimensionWeek 3 Introduction to differentiation and rectifiabilityWeek 4 To be determined by audience interest


Today

1 Agree some notation and terminology2 Agree a list of ‘test’ examples3 What properties should dimension have?4 Define some dimensions5 Test the definitions against (some of) our examples


Our base assumptions

We shall usually work in Rn with the Euclidean metric. (Andnearly always will specialise to the plane.)Most of the ideas however make sense in complete separablemetric spaces.


Part 1: Examples of sets


Simple sets

1 ∅2 finite set: {a1, . . . ,an}3 {0} ∪

{1n : n ∈ N

}4 traditional geometric shapes (line segment, triangle,

square,. . . )


Cantor setsDefinition

13 -Cantor set =

⋂Ei .

Some properties• Uncountable, compact, totally disconnected (and perfect —

no isolated points)• Has zero length (whatever that means)


Cantor set


Cantor sets in the plane


von Koch curve

Definition

Some properties• Has infinite length• Has zero area


von Koch curve


Sierpinski gasket

Has area zero and is compact (and perfect). An example of auniversal set — contains a homeomorphic image of everycompact planar curve .


A weird set

E =⋃

p,q∈Q, r∈Q+

S((p,q), r).

(Where S((p,q), r) denotes the circle centre (p,q) with radius r .)

This set is dense in the plane and has area zero.Can we talk about tangents for this set?


Besicovitch and Kakeya sets

1 What is the area of the smallest shape within which it ispossible to rotate a unit needle so that it is facing theopposite way.

2 What is the area of the smallest shape that contains a linesegment in every direction.


Besicovitch and Kakeya sets

1 For each ε > 0, there is a plane set E with area at most εinside which a unit segment may be moved continuously tolie in its original position but rotated through π. (Kakeyasets)

2 There is a plane set of area zero that contains a unitsegment in every direction. (Besicovitch set)


Perron trees

By Perrontree2.jpg: Jazzamderivative work: Mikhail Ryazanov - This file was derived from Perrontree2.jpg:, Public Domain,

https://commons.wikimedia.org/w/index.php?curid=24335166


https://commons.wikimedia.org/w/index.php?curid=24335166

Part 2: Dimension and measures


Properties expected of dimension• points are zero-dimensional• line segments are one-dimensional• squares (and other such ‘nice’ shapes) are two-dimensional• monotonicity: if E ⊆ F , then dim(E) ≤ dim(F ).• finite stability: dim(E ∪ F ) = max{dim(E), dim(F )}• Also nice to have countable stability:

dim

(∞⋃

i=1

Ei

)= sup{dim(Ei) : i = 1,2, . . .}.


box dimension

• Suppose that F is a (non-empty) bounded set in R2.• Fix a scale r > 0.• Consider the (almost disjoint) squares of side r , parallel to

the axes that cover the plane.


box dimension

• Let Nr (F ) be the number of squares from this grid that meetF .

Then r 7→ Nr (F ) is a well-defined, monotonic decreasingfunction of r for r > 0.


box dimension: line segment


box dimension: square


box dimension: formal definition

Upper box dimension

dimB(F ) = lim supr↘0

log Nr (F )

− log r.

Lower box dimension

dimB(F ) = lim infr↘0

log Nr (F )

− log r.

If both limits exist and have a common value, then this is the boxdimension, dimB(F ).


box dimension: Cantor set


box dimension: properties

• dim(E) ≤ dim(F ) if E ⊆ F• Gives right value to sets such as line segments, smooth

curves.• translations, rescalings, rotations and reflections do not

change box dimension• upper box dimension is finitely stable:

dimB(E ∪ F ) = max{dimB(E), dimB(F )}.

• lends itself to numerical estimation• dimB(F ) = dimB(clos(F ))

The last item is a problem: dimB(Q ∩ [0,1]) = 1.


Estimating length: 1

Estimating length: spiral

Spiral



Estimating length: any coverIf we estimate length by summing diameters of any cover:



Estimating lengthWe get a better estimate of length by summing diameters ofcovering sets whose diameter is at most r , where r is small:


definition: r -covers

r -coverLet r > 0. A countable collection of sets {Ui : i ∈ N} in Rn is anr -cover of E ⊆ Rn if

1 E ⊆⋃

i∈N Ui

2 diam(Ui) ≤ r for each i ∈ N.


Interlude: some notation

For a set A and s ≥ 0, we define

|A|s =

0 if A= ∅,1 if A 6= ∅ and s = 0,diam(A)s otherwise.


Hausdorff measures

Definition (s-dimensional Hausdorff measure)Suppose that F is a subset of Rn and s ≥ 0. For any r > 0, wedefine

Hsr (F ) = inf

{∞∑

i=1

|Ui |s : {Ui} is an r -cover of F

}.

The s-dimensional Hausdorff measure is then given by

Hs(F ) = limr↘0Hs

r (F ).

(It is possible to show that s-dimensional Hausdorff measure isin fact a measure — we shall do this later.1.)

1small print: Borel regular outer measureTCON (Open University) An introduction to GMT, part 1 October 2016 31 / 56

Simple examples1 For any s ≥ 0, Hs(∅) = 0.2 For any point x ∈ Rn,

Hs({x}) =

{1 if s = 0,0 if s > 0.

3 If A is a countable set, then Hs(A) = 0 unless s = 0.


Properties of Hausdorff measure

1 If s < t and Hs(F ) <∞, then Ht(F ) = 0. (So for t > n,Ht(Rn) = 0.)

2 If s is a non-negative integer, then Hs is a constant multipleof the usual s-dimensional volume. (counting measure,length measure, area, volume etc.) — we shall not provethis.


Properties of Hausdorff measure

1 If 0 < r < inf{d(x , y) : x ∈ E , y ∈ F}, then

Hsr (E ∪ F ) = Hs

r (E) +Hsr (F )

and so, in this case,

Hs(E ∪ F ) = Hs(E) +Hs(F ).

2 Let F ⊆ Rn and suppose that f : F → Rm is a mapping suchthat for some fixed constants c and α

|f (x)− f (y)| ≤ c|x − y |α whenever x , y ∈ F .

Then for each s, Hs/α(f (F )) ≤ cs/αHs(F ).


Hausdorff dimension

Definition (Hausdorff dimension)For a set F , we define the Hausdorff dimension of F by

dimH(F ) = inf{s ≥ 0 : Hs(F ) = 0} = sup{s : Hs(F ) =∞}.

Observations1 Hausdorff dimension is defined for any set F (unlike box

dimension).2 dimH(∅) = 03 If A is a countable set, then dimH(A) = 0. In particular, Q

has Hausdorff dimension 0.


Properties of Hausdorff dimension

1 If A ⊆ B, then dimH(A) ≤ dimH(B).2 If F1,F2,F3, . . . is a (countable) sequence of sets, then

dimH

(∞⋃

i=1

Fi

)= sup{dimH(Fi) : 1 ≤ i ≤ ∞}.

3 for bounded sets F , dimH(F ) ≤ dimB(F ).


Another simple example

Let F be the compact set formed as the limit of the followingconstruction.

So we split each square up into 3× 3 subsquares and removethe 2× 2 subsquares at the top right. What is dimH(F )?


Heuristic for finding what the dimension could beAssume that F has positive and finite s-dimensional Hausdorffmeasure when s = dimH(F ) and then represent F as a finitedisjoint union of scaled copies of F , Fi , say where Fi is a copy ofF scaled by λi . Then

Hs(F ) = Hs

(⋃i

Fi

)=∑

i

Hs(Fi) =∑

i

λsiHs(F ).

Dividing through by Hs(F ) then gives

1 =∑

i

λsi

and solving this for s gives the expected Hausdorff dimension,log 5/ log 3.


Covers provide an upper bound

Can use box dimension to find an upper bound, since

dimH(F ) ≤ dimB(F ).


Upper bound for our example.



Digression: minimal measure theory

(X ,d), a metric space. (Usually complete and separable)

Definition: (outer) measuresA set function µ : {A : A ⊆ X} → [0,∞] is a measure if

1 µ(∅) = 02 if A ⊆ B, then µ(A) ≤ µ(B)3 µ (

⋃∞i=1 Ai) ≤

∑∞i=1 µ(Ai)

Definition: measurable setA ⊆ X is measurable if

µ(E) = µ(E ∩ A) + µ(E \ A) for each E ⊆ X


Basic results

TheoremIf µ is a measure andM is the µ-measurable sets, then

1 M is a σ-algebra. [∅ ∈ M, closed under complements andcountable unions]

2 if µ(A) = 0, then A ∈M3 if A1,A2, . . . ∈M are pairwise disjoint, thenµ (⋃

Ai) =∑∞

i=1 µ(Ai)

4 if A1,A2, . . . ∈M, then1 µ (

⋃Ai) = limi→∞ µ(Ai), provided A1 ⊆ A2 ⊆ · · ·

2 µ (⋂

Ai) = limi→∞ µ(Ai), provided A1 ⊇ A2 ⊇ · · · andµ(A1) <∞


Regular measuresµ is a regular measure if for each A ⊆ X , there is aµ-measurable set B with A ⊆ B and µ(A) = µ(B).

LemmaSuppose that µ is a regular measure.If A1 ⊆ A2 ⊆ · · · , then

µ

(∞⋃

i=1

Ai

)= lim

i→∞µ(Ai).


Identifying measurable sets

DefinitionThe family of Borel sets in a metric space X is the smallestσ-algebra that contains the open subsets of X .

DefinitionA measure µ is:

1 a Borel measure if the Borel sets are µ-measurable2 Borel regular if it is a Borel measure and for each A ⊆ X ,

there is a Borel set B with A ⊆ B and µ(A) = µ(B).


TheoremLet µ be a measure on X. Then µ is a Borel measure if, andonly if,

µ(A ∪ B) = µ(A) + µ(B),

whenever inf{d(x , y) : x ∈ A, y ∈ B} > 0.

TheoremHs is a Borel regular measure for each s ≥ 0.


Definition (support)The support of a Borel measure µ is defined to be the smallestclosed set F for which µ(X \ F ) = 0.spt(µ) = X \

⋃{U : U is open and µ(U) = 0}.

Definition (Mass distribution)A Borel measure µ is a mass distribution on the set F if thesupport of µ is contained in F and 0 < µ(F ) <∞.


Mass distribution principleSuppose that µ is a mass distribution on a set F and for somereal number s, there are c > 0 and r0 > 0 so that

µ(U) ≤ c|U|s,

whenever U is a set with |U| ≤ r0. Then Hs(F ) ≥ µ(F )/c > 0and so dimH(F ) ≥ s.


lower bound for our example

Plan1 Use squares in construction of F and Proposition 1.7 to

define a mass distribution on F .2 Show hypothesis of Mass Distribution Principle is satisfied.3 Profit! (Or, at the least, deduce a lower bound on Hausdorff

dimension of F .)


lower bound for F


Packing measures

Definition (packing){(xi , ri) : i = 1,2, . . . ,n} is an r -packing of E if

1 xi ∈ E for each i2 0 < ri ≤ r for each i3 d(xi , xj) ≥ max{ri , rj} for i 6= j


Definition (packing measure)Let s ≥ 0 be given and let E ⊆ X . For r > 0, define

Psr (E) = sup

{n∑

i=1

r si : {(xi , ri)}n

i=1 is an r -packing of E

}

andPs

0(E) = limr↘0Ps

r (E).

The s-packing measure of E is given by

Ps(E) = inf

{∞∑

i=1

Ps0(Ei) : E ⊆

∞⋃i=1

Ei

}.

This defines a Borel regular measure.


Definition (packing dimension)For F ⊆ Rn, the packing dimension of F is given by

dimP(F ) = sup{s : Ps(F ) = +∞}.

Fortunately in Rn. . .

TheoremIf F ⊆ Rn, then the packing dimension of F is given by

dimP(F ) = inf

{sup

idimB(Fi) : F ⊆

∞⋃i=1

Fi , Fi bounded

}.


Usage

• box commonly used in analysis of real-world phenomena• Hausdorff dimension commonly used/investigated in pure

mathematics• packing dimension turns out to be important in the study of

fine properties of probabilistic phenomena such asBrownian motion.


Some references

K. J. Falconer, The geometry of fractal sets, CambridgeUniversity Press, 1985.

K. J. Falconer, Fractal geometry, Wiley 2nd Edition, 2003.

P. Mattila, Geometry of sets and measures in Euclidean spaces,Cambridge University Press, 1995.


Problems

Homework, if interested.1 Find an example of a compact set E ⊂ R for which

dimB(E) < dimB(E).

2 Determine the Hausdorff, packing and box dimensions of⋃p,q∈Q, r∈Q+

S((p,q), r).

(Where S((p,q), r) denotes the circle centre (p,q) withradius r .)


Documents

An introduction to Geometric Measure Theory Part 1 ...mcs.open.ac.uk/tcon2/teaching/LTCC-2016-week-1.pdfAn introduction to Geometric Measure Theory Part 1: dimension 3 October 2016