MATH 595: GEOMETRIC MEASURE 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 geometry and topology, and services a variety of areas: partial differential equations and the calculus of variations, geometric function theory, number 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 \fractal" 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, harmonic 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 manifolds, 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 fractals, necessitate extensions to non-Riemannian and nonsmooth spaces. Sub-Riemannian spaces, particularly, the sub-Riemannian Heisenberg group, 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 metric space 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) 8x; y; z 2 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 Euclidean space R equipped with the Euclidean metric dE: q 2 2 dE(x; y) = jx − yj; j(x1; : : : ; xn)j = x1 + ··· + xn: The open ball in X with center x 2 X and radius r 2 (0; 1) is B(x; r) = fy 2 X : d(x; y) < rg: The closed ball with this center and radius is B(x; r) = fy 2 X : d(x; y) ≤ rg: 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) closure B(x; r) of B(x; r) need not coincide with B(x; r). The diameter of a (nonempty) subset A ⊂ X is diam(A) := supfd(x; y): x; y 2 Ag and the distance between two (nonempty) sets A; B ⊂ X is dist(A; B) = inffd(x; y): x 2 A; y 2 Bg: We abbreviate dist(fxg;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) := fx 2 X : dist(x; A) < g = B(x; ): x2A 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; 1] (where P(X) = fA : A ⊂ Xg 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 n A) 8E ⊂ 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; · · · 2 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 \outer measure", 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 subsets 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;::: 2 Mµ(X). S1 (i) If A1 ⊂ A2 ⊂ A3 ⊂ · · · , then µ( i=1 Ai) = limi!1 µ(Ai). T1 (ii) If A1 ⊃ A2 ⊃ A3 ⊃ · · · , and µ(A1) < 1, then µ( i=1 Ai) = limi!1 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) < 1; • locally finite if for every x 2 X there exists r > 0 so that µ(B(x; r)) < 1; S1 • σ-finite if there exist A1;A2;::: ⊂ X so that X = i=1 Ai and µ(Ai) < 1 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 Borel set B with µ(A) = µ(B); • Radon if µ(K) < 1 for all compact sets K, µ(V ) = supfµ(K): K ⊂ V compactg for all open sets V , and µ(A) = inffµ(V ): V ⊃ A openg for all sets A; • doubling if it is locally finite, Borel, and there exists a constant Cµ < 1 so that 0 < µ(B(x; 2r)) ≤ Cµµ(B(x; r)) < 1 for all x 2 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 support spt µ of a Borel measure µ is the complement of the largest open set whose measure is zero. In other words, \ spt µ := fF : F ⊂ X is closed, µ(X n F ) = 0g: Exercise 1.2.5. Show that x 2 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).