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Home , Borel measure, Dirac measure, Finite measure, Geometric measure theory, Haar measure, Hausdorff measure

MATH 595: GEOMETRIC THEORY

FALL 2015

0. Introduction Geometric measure theory considers the structure of Borel sets and Borel measures in metric spaces. It lies at the border between differential and , and services a variety of : partial differential equations and the of variations, geometric function theory, , etc. The emphasis is on sets with a fine, irregular structure which cannot be well described by the classical tools of geometric analysis. Mandelbrot introduced the term “” to describe sets of this nature. Dynamical systems provide a rich source of examples: Julia sets for rational maps of one complex variable, limit sets of Kleinian groups, attractors of iterated function systems and nonlinear differential systems, and so on. In addition to its intrinsic interest, geometric measure theory has been a valuable tool for problems arising from real and complex analysis, , PDE, and other fields. For instance, rectifiability criteria and metric curvature conditions played a key role in Tolsa’s resolution of the longstanding Painlev´eproblem on removable sets for bounded analytic functions. Major topics within geometric measure theory which we will discuss include Hausdorff measure and dimension, density theorems, energy and capacity methods, almost sure di- mension distortion theorems, Sobolev spaces, tangent measures, and rectifiability. Rectifiable sets and measures provide a rich measure-theoretic generalization of smooth differential submanifolds and their volume measures. The theory of rectifiable sets can be viewed as an extension of differential geometry in which the basic machinery and tools of the subject (tangent spaces, differential operators, vector bundles) are replaced by approxi- mate, measure-theoretic analogs. In contrast with smooth , rectifiable sets form a complete system; limits of rectifiable sets (in a suitable sense) are rectifiable. Classically, geometric measure theory was developed in the setting of finite-dimensional Euclidean spaces. Contemporary trends in analysis and geometry in metric measure spaces, including analysis on , necessitate extensions to non-Riemannian and nonsmooth spaces. Sub-Riemannian spaces, particularly, the sub-Riemannian Heisenberg , are an important testing ground and model for the general theory. In the first part of the course we will give an extended introduction to geometric measure theory in Euclidean spaces. In the second part we will discuss ongoing efforts to extend this theory into non-Riemannian and general metric spaces. Motivation for such efforts will be indicated. We will emphasize notions of rectifiability and almost sure dimension distortion theorems in sub-Riemannian spaces. The notation [M] refers to the book Geometry of Sets and Measures in Euclidean Spaces by Pertti Mattila (Cambridge University Press, 1995).

1 2 Math 595: Geometric Measure Theory (Fall 2015)

1. Review of Measure Theory 1.1. Metric spaces. A is a set X together with a nonnegative, symmetric function d on X × X which satisfies the triangle inequality d(x, y) ≤ d(x, z) + d(z, y) ∀x, y, z ∈ X and vanishes only on the diagonal of X × X. We write (X, d) for the space X equipped with the metric d. Often we will not explicitly identify the metric d under consideration. Our n standard example is the R equipped with the Euclidean metric dE: q 2 2 dE(x, y) = |x − y|, |(x1, . . . , xn)| = x1 + ··· + xn. The open ball in X with center x ∈ X and radius r ∈ (0, ∞) is B(x, r) = {y ∈ X : d(x, y) < r}. The closed ball with this center and radius is B(x, r) = {y ∈ X : d(x, y) ≤ r}. Every metric space (X, d) is equipped with a canonical topology, obtained by taking the family of open balls as a base. In contrast with the situation in Euclidean space, • the center and radius of a ball in a general metric space are not uniquely determined. • the (topological) B(x, r) of B(x, r) need not coincide with B(x, r). The diameter of a (nonempty) A ⊂ X is diam(A) := sup{d(x, y): x, y ∈ A} and the distance between two (nonempty) sets A, B ⊂ X is dist(A, B) = inf{d(x, y): x ∈ A, y ∈ B}. We abbreviate dist({x},B) = dist(x, B). Observe that diam(A) = diam(A) and dist(A, B) = dist(A, B) for any A, B ⊂ X, where A denotes the closure of A. The diameter of any ball in a metric space satisfies diam B(x, r) ≤ 2r. Equality holds in Rn for all x and all r > 0, but need not hold in general metric spaces. For  > 0 and A ⊂ X the -neighborhood of A in X is the set [ N(A) := {x ∈ X : dist(x, A) < } = B(x, ). x∈A 1.2. Measures. For us, a measure on a space X will mean a nonnegative (possibly infinite- valued), monotonic, countably subadditive set function defined on the power set of X which vanishes on the empty set. More precisely, it is a function µ : P(X) → [0, ∞] (where P(X) = {A : A ⊂ X} denotes the power set of X) such that µ(∅) = 0, µ(A) ≤ µ(B) if A ⊂ B ⊂ X, and [ X µ( Ai) ≤ µ(Ai) i i if A1,A2, · · · ⊂ X. Math 595: Geometric Measure Theory (Fall 2015) 3

Recall that a set A ⊂ X is called (µ-)measurable if µ(E) = µ(E ∩ A) + µ(E \ A) ∀E ⊂ X.

In other words, A splits the measure of each subset of E. The collection Mµ(X) of µ- measurable sets forms a σ-algebra: it is closed under complements and countable unions. Each measure is countably additive on pairwise disjoint measurable sets: if A1,A2, · · · ∈ Mµ(X) and Ai ∩ Aj = ∅ whenever i 6= j, then [ X (1.2.1) µ( Ai) = µ(Ai). i i Remark 1.2.2. What we call a measure here is what is usually called an “”, with the term “measure” reserved for the restriction of the outer measure to the σ-algebra of measurable sets. The terminology which we use here is traditional in geometric measure theory. It is essentially equivalent; any countable additive nonnegative set function defined on a σ-algebra of of X can be extended to a measure on X (in the above sense). See Exercise 1.1 in [M]. Among various properties of measures and measurable sets we highlight the following.

Proposition 1.2.3. Let A1,A2,... ∈ Mµ(X). S∞ (i) If A1 ⊂ A2 ⊂ A3 ⊂ · · · , then µ( i=1 Ai) = limi→∞ µ(Ai). T∞ (ii) If A1 ⊃ A2 ⊃ A3 ⊃ · · · , and µ(A1) < ∞, then µ( i=1 Ai) = limi→∞ µ(Ai). From now on we assume that the underlying space X is equipped with a metric d. The Borel σ-algebra of X is the smallest σ-algebra containing the open sets. Elements of the Borel σ-algebra are called Borel sets. A measure µ on (X, d) is said to be • finite if µ(X) < ∞; • locally finite if for every x ∈ X there exists r > 0 so that µ(B(x, r)) < ∞; S∞ • σ-finite if there exist A1,A2,... ⊂ X so that X = i=1 Ai and µ(Ai) < ∞ for all i, • Borel if every Borel set is measurable; • Borel regular if it is Borel and each set A is a subset of a B with µ(A) = µ(B); • Radon if µ(K) < ∞ for all compact sets K, µ(V ) = sup{µ(K): K ⊂ V compact} for all open sets V , and µ(A) = inf{µ(V ): V ⊃ A open} for all sets A; • doubling if it is locally finite, Borel, and there exists a constant Cµ < ∞ so that

0 < µ(B(x, 2r)) ≤ Cµµ(B(x, r)) < ∞ for all x ∈ X and r > 0. All of the measures we will consider in this course will be Borel, and virtually all of them will be Borel regular. Although many measures which we will consider will be locally finite, a notable exception is the family of Hausdorff measures which will figure prominently in the course (see subsection 1.4 for the definition). The Hausdorff measure Hα on Rn is not locally finite for any 0 ≤ α < n. 4 Math 595: Geometric Measure Theory (Fall 2015)

All locally finite Borel regular measures on complete, separable metric spaces are Radon. Doubling measures are extensively studied in harmonic and geometric analysis. We will consider them in more detail in section 2. A necessary and sufficient condition for a measure µ on a metric space (X, d) to be Borel is Carath´eodory’s criterion: (1.2.4) µ(A ∪ B) = µ(A) + µ(B) whenever A, B ⊂ X satisfy dist(A, B) > 0. The spt µ of a µ is the complement of the largest whose measure is zero. In other words, \ spt µ := {F : F ⊂ X is closed, µ(X \ F ) = 0}. Exercise 1.2.5. Show that x ∈ spt µ if and only if µ(B(x, r)) > 0 for all r > 0. It is time to discuss examples. Example 1.2.6 (Counting measure). Let X be a set. For each subset A ⊂ X let µ(A) be the cardinality of A. Then µ is a measure in X. It is known as counting measure. Whenever X is equipped with a metric, counting measure is Borel regular. It is finite if and only if X is a finite set. It is locally finite if and only if X is a discrete set (every compact subset of X is finite). It is σ-finite if and only if X is countable. Example 1.2.7 (Dirac measures). Let X be a set. For fixed a ∈ X, let µ(A) = 1 if a ∈ A and µ(A) = 0 otherwise. Then µ is a finite measure. It is known as the (or point mass) at a, and is typically denoted δa. It is a whenever X is equipped with a metric. The following example demonstrates some of the subtlety associated to the notion of the support of a measure.

Example 1.2.8. Let q1, q2,... be an enumeration of the rational numbers and define ∞ X −i µ := 2 δqi . i=1 Then µ is a finite measure on R, and µ(R \ Q) = 0. However, spt µ = R. Example 1.2.9 (). The Lebesgue measure Ln in the Euclidean space Rn is defined as follows. For a rectangular parallelpiped Q = [a1, b1] × · · · × [an, bn] we set Qn Vol(Q) = j=1(bj − aj). Then ∞ n X L (A) := inf Vol(Qi), i=1 where the infimum is taken over all countable collections of rectangular parallelpipeds {Qi : S∞ n i = 1, 2,...} such that A ⊂ i=1 Qi. (If A is the empty set we declare L (A) = 0.) Lebesgue measure is a σ-finite doubling Radon measure. Moreover, Ln(Q) = Vol(Q) for each parallelpiped Q. To see that Lebesgue measure is doubling, it suffices to observe that n n n L (B(x, r)) = L (B(x, r)) = Ωnr n for all balls B(x, r) in R , where Ωn denotes a constant depending only on the dimension n. n (In fact, Ωn is the volume of the unit ball of R .) Math 595: Geometric Measure Theory (Fall 2015) 5

Example 1.2.10 (Restrictions of measures). Let µ be a measure on a set X and let Y be a fixed subset of X. The restriction of µ to Y , written µ Y , is the measure defined by (µ Y )(A) = µ(Y ∩ A), A ⊂ X. Example 1.2.11 (Pushforward measures). Let f : X → Y be a function and let µ be a measure on X. The pushforward of µ by f, written f#µ, is the measure on Y defined by −1 f#µ(A) = µ(f (A)) for A ⊂ Y . If µ is a Borel measure and f is a Borel function, then f#µ is a Borel measure. See [GMT, pp. 16–17] for a proof of the following facts: • If µ is a Radon measure on X with compact support and f : X → Y is continuous, then f#µ is a Radon measure on Y . • Let X and Y be compact metric spaces and let f : X → Y be a continuous surjection. For any Radon measure ν on Y there exists a Radon measure µ on X so that f#µ = ν.

Example 1.2.12. Let V be an m-dimensional subspace of Rn. Then V can be identified with Rm by fixing a basis of V . More precisely, there exists a linear bijection f : Rm → V . Via f we may push the Lebesgue measure from Rk to a measure on V . Thus we consider the m measure f#L on V . We will continue to call this the Lebesgue m-measure and will continue to denote it by Lm. Example 1.2.13. Let G be a metric group (i.e., a group equipped with a metric d : G×G → [0, ∞) such that the map (x, y) → xy−1 is continuous). A measure µ on G is left invariant if µ(gA) = µ(A) for all A ⊂ G, where gA = {gx : x ∈ A}. A measure µ is called a left Haar measure if it is a left invariant Radon measure. Right invariant measures and Haar measures are defined similarly. A measure is invariant if it is both left and right invariant, and is a Haar measure if it is both a left and a right Haar measure. Every locally compact metric group supports a left Haar measure. Moreover, any two left Haar measures are proportional, i.e., if µ and ν are left Haar measures on a locally compact metric group G, then µ = cν for some constant c > 0. If the group G is compact, it supports a finite Haar measure µ which is again unique up to constant multiples. In the latter case, it is traditional to require that µ be a , i.e., µ(G) = 1. Haar measure can constructed in arbitrary (locally) compact topological groups. There are some subtleties in the theory of Haar measures on topological groups which are not present in the metric setting. All of the examples we will consider in this course will in fact be metric groups. A property P of elements of X is said to hold µ-almost everywhere (µ-a.e.) if the set S of points x ∈ X at which property P does not hold satisfies µ(S) = 0. For instance, L1-a.e. is irrational.

1.3. Integration in measure spaces. Integrals1 Z Z f dµ = f(x) dµx A A of a µ- f : X → [−∞, ∞] over a measurable set A with respect to a measure µ are defined in the standard way. Recall that such are defined first

1We adopt a notational convention from [M], writing the element of integration dµx without any parentheses. 6 Math 595: Geometric Measure Theory (Fall 2015) for non-negative (extended real-valued) functions f : X → [0, ∞] and then extended to all functions by the rule Z Z Z (1.3.1) f dµ := f + dµ − f − dµ, A A A where f + = max{f, 0} and f − = max{−f, 0}; note that f = f + − f − and |f| = f + + f −. R + R − The value in (1.3.1) is well-defined only in case either A f dµ or A f dµ is finite. The space L1(µ) of µ-integrable functions consists of all µ-measurable functions f such that R A |f| dµ < ∞. When A = X we often omit the subscript on the sign. We abbreviate R f = R f(x) dx for integrals with respect to Lebesgue measure Ln. Rn Rn For 1 ≤ p < ∞ we introduce the Lp space Lp(µ) to be the space of µ-measurable functions f such that R |f|p dµ < ∞. Also L∞(µ) denotes the space of µ-essentially bounded functions; f ∈ L∞(µ) if there exists M < ∞ so that |f| ≤ M µ-a.e. The following result is an easy consequence of the definitions. It relates integrals with respect to a measure µ and its pushforward under a Borel map f : X → Y . Theorem 1.3.2. Let f : X → Y be a Borel mapping, let µ be a Borel measure on X, and R R let g be a non-negative Borel function on Y . Then Y g df#µ = X (g ◦ f) dµ. We will make use of many basic properties of integrals in measure spaces such as conver- gence theorems, Fubini’s theorem, H¨older’sinequality, and so on. For statements and proofs of these results, please consult any standard reference on real analysis and measure theory. 1.4. Carath´eodory’s construction and Hausdorff measures. We present Carath´eodory’s method for constructing Borel measures in metric spaces. Definition 1.4.1. Let (X, d) be a separable metric space, let F be a family of Borel subsets of X, and let h : [0, ∞) → [0, ∞) be a non-decreasing continuous function with h(0) = 0. For δ > 0 and A ⊂ X, let

h X Hδ (A) = inf h(diam Ei), i where the infimum is taken over all coverings E1,E2,... of A by elements from the class F h with diameter at most δ. (When A = ∅ we declare Hδ (∅) = 0.) h It is clear that Hδ (A) increases as δ & 0 (for fixed h and A). Set h h h H (A) = lim Hδ (A) = sup Hδ (A). δ&0 δ>0 h In order for the preceding definition of Hδ to be reasonable, we need to impose the following assumption on the family F: for each δ > 0, X can be covered by countably many elements of F each with diameter less than δ. This assumption is typically easy to verify. Standard families F which we consider include the full power set P(X), the family of all balls B(x, r), x ∈ X, r > 0, or (in the case X = Rn) the family of all rectangular parallelpipeds. h It is easy to check that Hδ is monotonic and countably subadditive, hence a measure on X. In rough terms, it measures the “size” of A from the perspective of efficient coverings of P A by sets of size at most δ, relative to the cost function i h(diam Ei). h h When δ > 0, Hδ is typically not additive on disjoint Borel sets. However, H is a Borel measure. This follows from the next proposition. Math 595: Geometric Measure Theory (Fall 2015) 7

Proposition 1.4.2. For any metric space (X, d), any family F, and any suitable function h, the measure Hh satisfies Carath´eodory’s criterion (1.2.4). Moreover, Hh is Borel regular. Proof. It suffices to prove that Hh(A ∪ B) ≥ Hh(A) + Hh(B) whenever A, B ⊂ X satisfy 1 dist(A, B) > 0. Fix 0 < δ < 3 dist(A, B). Suppose that A ∪ B is covered by a countable collection of sets E1,E2,... from F, with diam Ei < δ for all i. Since diam Ei < δ and dist(A, B) > 3δ, none of the sets Ei can intersect both A and B. Hence ∞ X X X h h h(diam Ei) ≥ h(diam Ei) + h(diam Ei) ≥ Hδ (A) + Hδ (B). i=1 i i Ei∩A6=∅ Ei∩B6=∅ h h h Taking the infimum over all such coverings of A ∪ B gives Hδ (A ∪ B) ≥ Hδ (A) + Hδ (B) and letting δ → 0 gives the desired inequality. The fact that Hh is Borel regular comes from the construction of Hh and the fact that the elements of F are Borel sets.  s Example 1.4.3. Let hs(t) = t for some 0 ≤ s < ∞, and let F be the class of all Borel subsets of X. The measure Hhs is called the s-dimensional Hausdorff measure on X. In this setting we write Hs in place of Hhs . Exercise 1.4.4. Show that H0 coincides with the counting measure on any metric space. Exercise 1.4.5. Show that H1 = L1 on the R. Exercise 1.4.6. For A ⊂ Rn, v ∈ Rn, r > 0 and s ≥ 0, show that Hs(A + v) = Hs(A) and Hs(rA) = rsHs(A). Here A + v = {a + v : a ∈ A} and rA = {ra : a ∈ A}. The following proposition is fundamental. Computing the exact proportionality constant is surprisingly difficult, relying on the isodiametric inequality for Rn. Some sources define the Hausdorff measures by including an additional multiplicative constant designed to make the measures Hn and Ln equal in each dimension n. We do not adopt this approach. Proposition 1.4.7. For each n, there exists c(n) > 0 so that Hn = c(n)Ln in Rn. Proof. Both Hn and Ln are Haar measures on Rn (viewed as an under vector addition). See Exercise 1.4.6. The conclusion follows from the uniqueness of Haar measures up to multiplicative constants.  The class of test sets F can be varied. For example, choosing F to be the set of all balls s in X and h = hs gives rise to the s-dimensional spherical Hausdorff measure S . It is not difficult to see that (1.4.8) Hs(A) ≤ Ss(A) ≤ 2sHs(A) for any A ⊂ X and any 0 ≤ s < ∞. Indeed, the left hand inequality follows immediately, s since the infimum in the definition of Sδ is taken over a restricted class of sets. To prove the right hand inequality, let δ > 0 and let A ⊂ ∪iEi for some collection of Borel sets E1,E2,... with diam Ei < δ. For each i choose any point xi ∈ Ei. Then Ei ⊂ Bi := B(xi, diam Ei) and diam Bi ≤ 2 diam Ei < 2δ. It follows that s X s s X s S2δ(A) ≤ (diam Bi) ≤ 2 (diam Ei) i i s s s s s and so S2δ(A) ≤ 2 Hδ(A) ≤ 2 H (A). Taking the limit as δ → 0 finishes the proof. 8 Math 595: Geometric Measure Theory (Fall 2015)

Another important variant in the Euclidean setting is the family of dyadic Hausdorff measures. First, we introduce the dyadic decomposition of Rn. For each integer m, we let Qm denote the family of half-open cubes  k k + 1  k k + 1 Q = 1 , 1 × · · · × n , n , 2m 2m 2m 2m n where k1, . . . , kn ∈ Z. For fixed m, the family Qm of all such cubes defines a partition of R −m into disjoint sets. We call elements of Qm dyadic cubes of scale 2 . Note that √ n diam Q = ∀ Q ∈ Q . 2m m n s For s ≥ 0, the dyadic s-dimensional Hausdorff measure in R , denoted Hdyadic, is the measure obtained by the preceding construction of Carath´eodory for h = hs, with F equal to the family of all dyadic cubes (of all scales). Coverings with dyadic cubes have an important advantage over coverings with balls or n more general sets. If a set A ⊂ R is covered by a collection Q1,Q2,... of dyadic cubes (not necessarily all of the same scale), then A is covered by a subcollection Qi1 ,Qi2 ,... which is pairwise disjoint. This fact can be deduced from the following property of dyadic cubes: any two dyadic cubes in Rn are either pairwise disjoint, or one is contained in the other. As was the case for spherical Hausdorff measures, dyadic Hausdorff measures on Rn are comparable to the corresponding Hausdorff measures (defined by arbitrary coverings). Proposition 1.4.9. For any A ⊂ Rn and s ≥ 0, s s n s s/2 s H (A) ≤ Hdyadic(A) ≤ 3 2 n H (A). Proof. The left hand inequality is trivial. For the right hand inequality, fix a set A ⊂ Rn, fix δ > 0, and consider a covering {Ei : i = 1, 2,...} of A by arbitrary sets with diam Ei < δ for all i. For each i, choose an integer mi so that 1 1 ≤ diam Ei < . 2mi+1 2mi n −mi We claim that Ei is covered by 3 dyadic cubes of scale 2 . To see this, note that if some −mi dyadic cube Q of scale 2 intersects Ei, then Ei is contained in the union of all dyadic cubes Q0 of scale 2−mi such that Q0 ∩ Q 6= ∅, and the number of such cubes is 3n. For the 0 latter claim, it suffices to note that if x ∈ Ei ∩ Q and y ∈ Ei is not in any such cube Q , then 1 ≤ |x − y| ≤ diam Ei 2mi n which contradicts the choice of mi. To each Ei we may therefore associate 3 dyadic cubes 0 −mi Q of scale 2 whose union contains Ei. The diameter of any of these cubes satisfies √ 0 n √ √ diam Q = ≤ 2 n diam Ei < 2 nδ. 2mi √ Let  := 2 nδ. Then s X X 0 s n √ s X s Hdyadic,(A) ≤ (diam Q ) ≤ 3 (2 n) (diam Ei) i Q0 i 0 Q assoc to Ei s n s s/2 s n s s/2 s and so Hdyadic,(A) ≤ 3 2 n Hδ(A) ≤ 3 2 n H (A). Letting δ (and hence also ) tend to zero yields the right hand inequality and completes the proof of the proposition.  Math 595: Geometric Measure Theory (Fall 2015) 9

1.5. Hausdorff dimension and behavior of Hausdorff measure/dimension under mappings. The following lemma describes how the Hausdorff measure changes as the ex- ponent s varies. It is critical for the definition of the Hausdorff dimension. Lemma 1.5.1. Let A ⊂ X and let 0 ≤ s < t < ∞. Then (i) Hs(A) < ∞ implies Ht(A) = 0, and (ii) Ht(A) > 0 implies Hs(A) = ∞. Of course, (i) and (ii) are equivalent. Taken together, they show that the value Hs(A) can be positive and finite for at most one value s. This value is the Hausdorff dimension of the set A.

Definition 1.5.2. The Hausdorff dimension of A ⊂ X is the unique value s0 ∈ [0, ∞) so s s that H (A) = 0 for all s > s0 and H (A) = ∞ for all 0 ≤ s < s0. We will write H dim(A) for the Hausdorff dimension of A. If necessary we write H dimd(A) to indicate the dependence of this value on the metric d. Here are some simple properties of this dimension: • monotonicity: H dim(A) ≤ H dim(B) if A ⊂ B; S • countable stability: H dim(A) = supi H dim(Ai) if A1,A2, · · · ⊂ X and A = i Ai; • H dim(A) = 0 for every countable set A; • H dim(A) = m whenever A is a smooth m-dimensional submanifold of Rn. Notice that the value of the Hausdorff dimension is unchanged if we replace the Hausdorff measures with the spherical Hausdorff measures. This follows from (1.4.8).

Definition 1.5.3. A map f : X → Y between metric spaces (X, dX ) and (Y, dY ) is said to be L-Lipschitz if dY (f(x), f(y)) ≤ LdX (x, y) for every x, y ∈ X. A homeomorphism f : X → Y is said to be L-bi-Lipschitz if 1 d (x, y) ≤ d (f(x), f(y)) ≤ Ld (x, y) L X Y X for every x, y ∈ X. A map f : X → Y is said to be an L-bi-Lipschitz embedding if it is an L-bi-Lipschitz homeomorphism from X onto its image f(X) ⊂ Y . The behavior of Hausdorff measures and dimensions under Lipschitz maps is easy to observe. Proposition 1.5.4. Let f :(X, d) → (Y, d0) be L-Lipschitz and let A ⊂ X. Then (i) Hs(f(A)) ≤ LsHs(A) for every 0 ≤ s < ∞; (ii) H dim f(A) ≤ H dim A. Corollary 1.5.5. Let f :(X, d) → (Y, d0) be bi-Lipschitz. Then H dim f(A) = H dim A for every A ⊂ X. Thus Hausdorff dimension is a bi-Lipschitz invariant of metric spaces. One of the central problems in analysis in metric spaces is to determine a list of metric properties or quantities which suffice to characterize spaces up to bi-Lipschitz equivalence (or up to equivalence by mappings in some other natural metrically defined class). 0 0 Exercise 1.5.6. Show that if f :(X, d) → (Y, d ) satisfies d (f(x1), f(x2)) = Ld(x1x2) for s s s all x1, x2 ∈ X, then H (f(A)) = L H (A) for all A ⊂ X and all s ≥ 0. 10 Math 595: Geometric Measure Theory (Fall 2015)

1.6. An example: the . To illustrate the computation of Hausdorff measure and dimension, we consider the classical Cantor set  1 2 1 2 7 8  C = [0, 1] \ ( , ) ∪ ( , ) ∪ ( , ) ∪ · · · . 3 3 9 9 9 9 1 We may write C = CL ∪ CR where CL and CR are similarity copies of C, scaled by ratio 3 . 1 1 2 More precisely, CL = 3 C and CR = 3 C + 3 (where the notation is as in Exercise 1.4.6). If s0 s0 denotes the Hausdorff dimension of C, and if we assume that 0 < H (C) < ∞, then 1 Hs0 (C) = Hs0 (C ) + Hs0 (C ) = 2( )s0 Hs0 (C) L R 3

1 s0 and we deduce that 2( 3 ) = 1 and log 2 (1.6.1) H dim(C) = s := . 0 log 3 In order to make this computation rigorous we must justify the italicized assumption. Here are the details. Consider finite words w in the letters L and R, i.e., w = w1w2 ··· wm with wi ∈ {L, R}. We call m the length of such a word, and write Wm for the set of all words of length m. Each word w ∈ Wm generates a subset Cw ⊂ C by the obvious iterative procedure, and [ C = Cw

w∈Wm −m for each m. Note that diam Cw = 3 for each w ∈ Wm. It follows that

s0 X s0 m −ms0 H3−m (C) ≤ (diam Cw) = 2 · 3 = 1. w∈Wm

s0 Letting m → ∞ gives H (C) ≤ 1 so H dim C ≤ s0. On the other hand, suppose that {Ei} is an arbitrary cover of C. By replacing Ei with its convex hull and expanding slightly if necessary, we may assume that the Ei’s are open intervals; since C is compact we may assume that there are only finitely many Ei’s. For each i, let mi be the integer such that

−mi−1 −mi (1.6.2) 3 ≤ diam Ei < 3 . 0 −m For distinct words w, w in Wm, we have dist(Cw,Cw0 ) ≥ 3 . It follows that each m−mi Ei can meet at most one set Cw, w ∈ Wmi , and so can meet at most 2 such sets Cw, m w ∈ Wm, m ≥ mi. Since all 2 sets Cw, w ∈ Wm are met, we have

m X m−mi s0 m X s0 2 ≤ 2 ≤ 3 · 2 (diam Ei) , i i where the second inequality comes from (1.6.2). Hence 1 Hs0 (C) ≥ Hs0 (C) ≥ 3−s0 = ∞ 2 and so H dim C ≥ s0. Now (1.6.1) is justified. Remark 1.6.3. Even in this simple example, the computation of lower bounds for Hausdorff measures was somewhat involved. In the next few lectures, we will discuss a variety of methods for obtaining lower bounds for Hausdorff measures. Math 595: Geometric Measure Theory (Fall 2015) 11

1.7. Hausdorff dimension vs. topological dimension. The small inductive dimension dim X of a X is defined as follows: the empty set is declared to have dimension −1, and a topological space X has dimension ≤ n if each point in X has arbitrarily small neighborhoods whose boundaries have dimension ≤ n − 1. For example, Rn has small inductive dimension n, while the Cantor set C has dimension zero. Remark 1.7.1. The preceding is only one of several definitions of dimension in topological spaces. Other possibilities include the (Lebesgue) covering dimension and the large inductive dimension. For separable metric spaces, all of these definitions are equivalent. We will always work in this setting; we call dim X the topological dimension of X. Observe that in contrast with Hausdorff dimension, the topological dimension is a purely topological notion; homeomorphically equivalent metric spaces have equal topological di- mension. The relation between these two notions is contained in the following theorem. Theorem 1.7.2. Let (X, d) be a metric space. Then H dim X ≥ dim X. More precisely, if X is a metrizable topological space, then dim X ≤ H dim X for any metric d on X which generates the topology. It can be shown that

(1.7.3) dim X = inf H dimd X d where the infimum is taken over all metrics d on X which generate the topology. We will not prove or use (1.7.3) in this course. Before proving Theorem 1.7.2, let us consider a special case. Proposition 1.7.4. Let (X, d) be a metric space with H dim X < 1. Then dim X = 0.

Proof of Proposition 1.7.4. Let x0 be a fixed point in X and consider the 1-Lipschitz function f on X given by f(x) = d(x, x0). By Proposition 1.5.4, H dim f(X) < 1. Thus L1(f(X)) = H1(f(X)) = 0; recall that L1 denotes the Lebesgue measure in R. It follows that f(X) contains no open neighborhood of 0 = f(x0), i.e., there exists rn → 0, rn ∈/ f(X). Then the “metric spheres”

S(x0, r) := {y ∈ X : d(x0, y) = rn}, n = 1, 2,..., are all empty. Since the boundary of B(x0, r) is contained in S(x0, r), we conclude that x0 has arbitrarily small neighborhoods with empty boundary. Since x0 was arbitrary the conclusion follows.  To prove Theorem 1.7.2, it suffices to show the following lemma. Lemma 1.7.5 can be viewed as a “Fubini-type” theorem for Hausdorff measures in metric spaces.

1+s s Lemma 1.7.5. Suppose that H (X) = 0 for some s ∈ Z, s ≥ −1. Then (i) H (∂B(x0, r)) = 0 for all x0 ∈ X and a.e. r > 0, and (ii) dim X ≤ s. Proof of Lemma 1.7.5. Assume first that (i) has been proved. We prove (ii) by induc- tion on s. The case s = −1 is obvious. Suppose that the lemma holds for some integer s − 1 and all metric spaces, and let (X, d) be a metric space such that H1+s(X) = 0. s By (i), H (∂B(x0, r)) = 0 for all x0 ∈ X and a.e. r > 0. By the inductive hypothesis, dim ∂B(x0, r) ≤ s − 1 for these x0 and r. Thus every x0 ∈ X has a neighborhood basis consisting of sets B(x0, r) whose boundary has dimension ≤ s − 1, so dim X ≤ s. 12 Math 595: Geometric Measure Theory (Fall 2015)

It remains to show that (i) holds. Fix x0 ∈ X and a ball B ⊂ X. Define r± by

r− = inf d(x0, x) = dist(x0,B) and r+ = sup d(x0, x). x∈B x∈B

Note that r+ − r− ≤ diam B by the triangle inequality. Since ∂B(x0, r) meets B only for r ∈ [r−, r+], we have Z ∞ Z r+ s s (diam ∂B(x0, r) ∩ B) dr = (diam ∂B(x0, r) ∩ B) dr 0 r− s 1+s ≤ (diam B) (r+ − r−) ≤ (diam B) . 1+s Since H (X) = 0 we can choose a sequence of balls B1,B2,... with diam Bi <  and X 1+s (diam Bi) < . i Hence Z ∞ X s  > (diam ∂B(x0, r) ∩ Bi) dr i 0 Z ∞ Z ∞ X s s ≥ (diam ∂B(x0, r) ∩ Bi) dr ≥ H (∂B(x0, r)) dr. 0 i 0 Bi∩∂B(x0,r)6=∅ R ∞ s s Letting  → 0 gives 0 H (∂B(x0, r)) dr = 0 so H (∂B(x0, r)) = 0 for a.e. r > 0.  Exercise 1.7.6. Let H be the family of all non-decreasing functions h : [0, ∞) → [0, ∞) such that h(0) = 0. Define a partial ordering h ≺ k on H by the requirement h(t)/k(t) → 0 as t → 0. Show that h ≺ k and Hk(X) < ∞ ⇒ Hh(X) = 0. Conclude that the generalized Hausdorff measure Hh(X) can be positive and finite for at most one such function h within any totally ordered subclass of H. s s Exercise 1.7.7. For any s > 0 and any (X, d), show that H (X) = 0 iff H∞(X) = 0. 1 Exercise 1.7.8. For any connected metric space (X, d), show that diam X ≤ H∞(X). 1.8. References for further reading. Our primary reference in this course is [M] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge studies in advanced mathematics, vol. 44, Cambridge University Press, 1995. Some of the material in this section can be found in Chapters 1 and 4 of [M]. Other good references for Hausdorff measures and their applications include [R] C. A. Rogers, Hausdorff measures, Cambridge Mathematical Library, Cam- bridge University Press, 1998. [F] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, second ed., Wiley and Sons, 2003. The classical reference for topological dimension theory is [HW] W. Hurewicz and H. Wallman, Dimension theory, Princeton Mathematical Series, vol. 4, Princeton University Press, 1941. Some of the material in subsection 1.7 is taken from [HW].