measure theory geometric introduction

Introduction to Geometric Measure Theory Urs Lang April 20, 2005 Abstract These are the notes to four one-hour lectures I delivered at the spring school “Geometric Measure Theory: Old and New” that took place in Les Diablerets, Switzerland, from April 3–8, 2005 (see http://igat.epfl.ch/diablerets05/). The first three of these lec- tures were intended to provide the fundamentals of the “old” theory of rectifiable sets and currents in euclidean space as developed by Besi- covitch, Federer–Fleming, and others. The fourth lecture, independent of the previous ones, discussed some metrique space techniques that are useful in connection with the new metric approach to currents by Ambrosio–Kirchheim. Other short courses were given by G. Alberti, M. Cs¨ornyei, B. Kirchheim, H. Pajot, and M. Z¨ahle. Contents Lecture 1: Rectifiability 3 Lipschitz maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Area formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Rectifiable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Lecture 2: Normal currents 14 Vectors, covectors, and forms . . . . . . . . . . . . . . . . . . . . . 14 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Normal currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Results for n-currents in R n . . . . . . . . . . . . . . . . . . . . . . 21 Lecture 3: Integral currents 22 Integer rectifiable currents . . . . . . . . . . . . . . . . . . . . . . . 22 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . 23 Minimizing currents . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1 Lecture 4: Some metric space techniques 27 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Gromov–Hausdorff convergence . . . . . . . . . . . . . . . . . . . . 28 Ultralimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References 34 2 Lecture 1: Rectifiability Lipschitz maps Let X, Y be metric spaces, and let λ ∈ [0, ∞). A map f : X → Y is λ- Lipschitz if d(f(x), f (x  )) ≤ λ d(x, x  ) for all x, x  ∈ X; f is Lipschitz if Lip(f) := inf{λ ∈ [0, ∞) : f is λ-Lipschitz} < ∞. The following basic extension result holds, see [McS] and the footnote in [Whit]. 1.1 Lemma (McShane, Whitney) Suppose X is a metric space and A ⊂ X. (1) For n ∈ N, every λ-Lipschitz map f : A → R n admits a √ nλ-Lipschitz extension ¯ f : X → R n . (2) For any set J, every λ-Lipschitz map f : A → l ∞ (J) has a λ-Lipschitz extension ¯ f : X → l ∞ (J). Proof : (1) For n = 1, put ¯ f(x) := inf{f(a) + λ d(a, x) : a ∈ A}. For n ≥ 2, f = (f 1 , . . . , f n ), extend each f i separately. (2) For f = (f j ) j∈J , extend each f j separately. ✷ In (1), the factor √ n cannot be replaced by a constant < n 1/4 , cf. [JohLS] and [Lan]. In particular, Lipschitz maps into a Hilbert s pace Y cannot b e extended in general. However, if X is itself a Hilbert space, one has again an optimal result: 1.2 Theorem (Kirszbraun, Valentine) If X, Y are Hilbert spaces, A ⊂ X, and f : A → Y is λ-Lipschitz, then f has a λ-Lipschitz extension ¯ f : X → Y . See [Kirs], [Val], or [Fed, 2.10.43]. A generalization to metric spaces with curvature bounds was given in [LanS]. The next result characterizes the extendability of partially defined Lips- chitz maps from R m into a complete metric space Y ; it is useful in connection with the definition of rectifiable sets (Def. 1.13). We call a metric space Y Lipschitz k-connected if there is a constant c ≥ 1 such that every λ-Lipschitz map f : S k → Y admits a cλ-Lipschitz extension ¯ f : B k+1 → Y ; here S k and B k+1 denote the unit sphere and closed ball in R k+1 , endowed with the in- duced metric. Every Banach space is Lipschitz k-connected for all k ≥ 0. The sphere S n is Lipschitz k-connected for k = 0, . . . , n − 1. 3 1.3 Theorem (Lipschitz maps on R m ) Let Y be a complete m etric space, and let m ∈ N. Then the following statements are equivalent: (1) Y is Lipschitz k-connected for k = 0, . . . , m − 1. (2) There is a constant c such that every λ-Lipschitz map f : A → Y , A ⊂ R m , has a cλ-Lipschitz extension ¯ f : R m → Y . The idea of the proof goes back to Whitney [Whit]. Compare [Alm1, Thm. (1.2)] and [JohLS]. Proof : It is clear that (2) implies (1). Now suppose that (1) holds, and let f : A → Y be a λ-Lipschitz map, A ⊂ R m . As Y is complete, assume w.l.o.g. that A is closed. A dyadic cube in R m is of the form x + [0, 2 k ] m for some k ∈ Z and x ∈ (2 k Z) m . Denote by C the family of all dyadic cubes C ⊂ R m \ A that are maximal (with respect to inclusion) subject to the condition diam C ≤ 2 d (A, C). They have pairwise disjoint interiors, cover R m \ A, and satisfy d(A, C) < 2 diam C, for otherwise the next bigger dyadic cube C  containing C would still fulfill diam C  = 2 diam C ≤ 2(d(A, C) − diam C) ≤ 2 d(A, C  ). Denote by Σ k ⊂ R m the k-skeleton of this cubical decomposition. Extend f to a Lipschitz map f 0 : A ∪Σ 0 → Y by precomposing f with a nearest point retraction A ∪ Σ 0 → A. Then, for k = 0, . . . , m − 1, successively extend f k to f k+1 : A ∪Σ k+1 → Y by means of the Lipschitz k-connectedness of Y . As A ∪ Σ m = R m , ¯ f := f m is the desired extension of f . ✷ Differentiability Recall the following definitions. 1.4 Definition (Gˆateaux and Fr´echet differential) Suppose X, Y are Banach spaces, f maps an open set U ⊂ X into Y , and x ∈ U. (1) The map f is Gˆateaux differentiable at x if the directional derivative D v f(x) exists for every v ∈ X and if there is a continuous linear map L: X → Y such that L(v) = D v f(x) for all v ∈ X. Then L is the Gˆateaux differential of f at x. 4 (2) The map f is (Fr´echet) differentiable at x if there is a continuous linear map L: X → Y such that lim v→0 f(x + v) − f(x) − L(v) v = 0. Then L =: Df x is the (Fr´echet) differential of f at x. The map f is Fr´echet differentiable at x iff f is Gˆateaux differentiable at x and the limit in L(u) = lim t→0 (f(x + tu) − f(x))/t exists uniformly for u in the unit sphere of X, i.e. for all  > 0 there is a δ > 0 such that f(x + tu) − f(x) − tL(u) ≤ |t| whenever |t| ≤ δ and u ∈ S(0, 1) ⊂ X. 1.5 Lemma (differentiable Lipschitz maps) Suppose Y is a Banach space, f : R m → Y is Lipschitz, x ∈ R m , D is a dense subset of S m−1 , D u f(x) exists for every u ∈ D, L: R m → Y is linear, and L(u) = D u f(x) for all u ∈ D. Then f is Fr´echet differentiable at x with differential Df x = L. In particular, if f : R m → Y is Lipschitz and Gˆateaux differentiable at x, then f is Fr´echet differentiable at x. Proof : Let  > 0. Choose a finite set D  ⊂ D such that for every u ∈ S m−1 there is a u  ∈ D  with |u − u  | ≤ . Then there is a δ > 0 such that f(x + tu  ) − f(x) − tL(u  ) ≤ |t| whenever |t| ≤ δ and u  ∈ D  . Given u ∈ S m−1 , pick u  ∈ D  with |u−u  | ≤ ; then f(x + tu) − f(x) − tL(u) ≤ |t|+ f(x + tu) − f(x + tu  ) + |t|L(u − u  ) ≤ (1 + Lip(f) + L)|t| for all |t| ≤ δ. ✷ 1.6 Theorem (Rademacher) Every Lipschitz map f : R m → R n is differentiable at L m -almost all points in R m . This was originally proved in [Rad]. 5 Proof : It suffices to prove the theorem for n = 1; in the general case, f = (f 1 , . . . , f n ) is differentiable at x iff each f i is differentiable at x. In the case m = 1 the function f : R → R is absolutely continuous and hence L 1 -almost everywhere differentiable. Now let m ≥ 2. For u ∈ S m−1 , denote by B u the set of all x ∈ R m where D u f(x) exists and by H u the linear hyperplane orthogonal to u. For x 0 ∈ H u , the function t → f(x 0 + tu) is L 1 -almost everywhere differentiable by the result for m = 1, hence H 1 ((x 0 + Ru) \ B u ) = 0. Since B u is a Borel set, Fubini’s theorem implies L m (R m \ B u ) = 0. Now choose a dense countable subset D of S m−1 . Then it follows that for L m -almost every x ∈ R m , D u f(x) and D e i f(x) exist for all u ∈ D and i = 1, . . . , m; in particular, the formal gradient ∇f(x) := (D e i f(x), . . . , D e m f(x)) exists. We show that for L m -almost all x ∈ R m we have, in addition, the usual relation D u f(x) = ∇f(x), u for all u ∈ D. The theorem then follows from Lemma 1.5. Let ϕ ∈ C ∞ c (R m ). By Lebesgue’s bounded convergence theorem, lim t→0+  f(x + tu) − f(x) t ϕ(x) dx =  D u f(x)ϕ(x) dx, lim t→0+  f(x) ϕ(x − tu) − ϕ(x) t dx = −  f(x)D u ϕ(x) dx. Substituting x + tu by x in the term f(x + tu)ϕ(x) we see that the two left-hand s ides coincide. Hence,  D u f(x)ϕ(x) dx = −  f(x)D u ϕ(x) dx, and s imilarly  ∇f(x), uϕ(x) dx = −  f(x)∇ϕ(x), udx. Now the right-hand sides of these two identities coincide. As ϕ ∈ C ∞ c (R m ) is arbitrary, we conclude that D u f(x) = ∇f(x), u for L m -almost every x ∈ R m . ✷ 6 1.7 Theorem (Stepanov) Every function f : R m → R n is differentiable at L m -almost all points in the set L(f) :=  x: lim sup y→x |f(y) − f(x)|/|y −x| < ∞  . This generalization of Rademacher’s theorem was proved in [Ste]. The following elegant argument is due to Mal´y [Mal]. Proof : It suffices to consider the case n = 1. Le t (U i ) i∈N be the family of all open balls in R m with rational center and radius such that f|U i is bounded. This family covers L(f). Let a i : U i → R be the supremum of all i-Lipschitz functions ≤ f|U i , and let b i : U i → R be the infimum of all i-Lipschitz functions ≥ f|U i . Note that a i , b i are i-Lipschitz and a i ≤ f|U i ≤ b i . Let A i := {x ∈ U i : both a i and b i are differentiable at x}. By Rademacher’s theorem, Z :=  ∞ i=1 U i \ A i has measure zero. Let x ∈ L(f) \ Z. We show that for some i, x ∈ A i and a i (x) = b i (x); then f is differentiable at x. Since x ∈ L(f), there is a radius r > 0 such that |f(y) − f(x)| ≤ λ|y − x| for all y ∈ B(x, r) and for some λ independent of y. Choose i such that i ≥ λ and x ∈ U i ⊂ B(x, r). Since x ∈ Z, x ∈ A i . By the definition of a i and b i , f(x) − i|y −x| ≤ a i (y) ≤ f(y) ≤ b i (y) ≤ f(x) + i|y −x| for all y ∈ U i . Hence, a i (x) = b i (x). ✷ Generalizations of these results to maps between Banach spaces or even more general classes of metric spaces are a topic of current research. Finally, we state Whitney’s extension theorem for C 1 functions and an application, cf. [Whit], [Fed, 3.1.14] and [Sim, 5.3], [Fed, 3.1.16]. 1.8 Theorem (Whitney) Suppose f : A → R is a function on a closed set A ⊂ R m , g : A → R m is continuous, and for all compact sets C ⊂ A and all  > 0 there is a δ > 0 such that |f(y) − f(x) − g(x), y −x| ≤ |y −x| whenever x, y ∈ C and |y−x| ≤ δ. Then there exists a C 1 function ¯ f : R m → R with ¯ f|A = f and ∇ ¯ f|A = g. 1.9 Theorem (C 1 approximation of Lipschitz functions) If f : R m → R is Lipschitz and  > 0, then there is a C 1 function ¯ f : R m → R such that L m ({x ∈ R m : f(x) = ¯ f(x)}) < . 7 Proof : By Rademacher’s theorem, f is almost everywhere differentiable, and g := ∇f is a measurable function. According to Lusin’s theorem, there is a closed set B ⊂ R m with L m (R m \B) < /2 such that g|B is continuous. For x ∈ B and i ∈ N, let r i (x) := sup |f(y) − f(x) − g(x), y −x|/ |y −x|, the supremum taken over all y ∈ B with 0 < |y − x| ≤ 1/i. We know that r i → 0 pointwise on B as i → ∞. By Egorov’s theorem, there is a closed set A ⊂ B with L m (B \ A) < /2 such that r i → 0 uniformly on compact subsets of A. Now extend f|A to R m by means of 1.8. ✷ Area formula The next goal is to prove Theorem 1.12 below. We start with a technical lemma, cf. [Fed, 3.2.2], [EvaG, p. 94]. 1.10 Lemma (Borel partition) Suppose f : R m → R n is Lipschitz, and B is the set of all x where Df x exists and has rank m. Then for every λ > 1 there exist a Borel partition (B i ) i∈N of B and a sequence of euclidean norms · i on R m (i.e. · i is induced by an inner product), such that λ −1 v i ≤ |Df x (v)| ≤ λv i , λ −1 y −x i ≤ |f(y) − f(x)| ≤ λy −x i for all x, y ∈ B i and v ∈ R m . Proof : Choose a sequence of euclidean norms  ·  j on R m such that for every euclidean norm · on R m and for every  > 0 there is a j ∈ N with (1 − )v j ≤ v ≤ (1 + )v j for all v ∈ R m . Given λ > 1, pick δ > 0 such that λ −1 + δ < 1 < λ −δ. For j, k ∈ N, denote by B jk the Borel set of all x ∈ B with (i) (λ −1 + δ)v j ≤ |Df x (v)| ≤ (λ −δ)v j for v ∈ R m , (ii) |f(x + v) − f(x) − Df x (v)| ≤ δv j for |v| ≤ 1/k. To see that the B jk cover B, let x ∈ B, choose j ∈ N such that (i) holds, let c j > 0 be such that |v| ≤ c j v j for all v ∈ R m , and pick k ∈ N such that |f(x + v) − f(x) − Df x (v)| ≤ (δ/c j )|v| whenever |v| ≤ 1/k; then x ∈ B jk . Now if C ⊂ B jk is a set with diam C ≤ 1/k, then |f(x + v) − f(x)| ≤ |Df x (v)|+ δv j ≤ λv j , |f(x + v) − f(x)| ≥ |Df x (v)|−δv j ≤ λ −1 v j 8 whenever x, x + v ∈ C. By subdividing and relabeling the sets B jk appro- priately we obtain the result. ✷ 1.11 Definition (jacobian) Let L: X → Y be a linear map between two inner product spaces, where dim X = m. The m-dimensional jacobian J m (L) of L is the number satis- fying J m (L) = H m (L(A))/H m (A) =  det(L ∗ ◦ L) for all A ⊂ X with H m (A) > 0, where L ∗ : Y → X is the adjoint map. If  ·  is a euclidean norm on R m , we write J m ( · ) for J m (L) where L: R m → (R m ,  · ) is the identity map. 1.12 Theorem (area formula) Suppose f : R m → R n is Lipschitz with m ≤ n. (1) If A ⊂ R m is L m -measurable, then  A J m (Df x ) dx =  R n #(f −1 {y} ∩A) dH m (y). (2) If u is an L m -integrable function, then  R m u(x)J m (Df x ) dx =  R n  x∈f −1 {y} u(x) dH m (y). Cf. [Fed, 3.2.3], [EvaG, Sect. 3.3]. The formula says, in particular, that the differential geometric volume of an injective C 1 map f : U → R n , U an open subset of R m , equals H m (f(U)). For a metric space version, see [Kir]. Proof : (1) We assume w.l.o.g. that L m (A) < ∞. Case 1: A ⊂ {x : Df x exists and has rank m}. Let λ > 1. Using Lemma 1.10 we find a measurable partition (A i ) i∈N of A and a sequence of euclidean norms  ·  i on R m such that f|A i is injective, λ −m H m · i (A i ) ≤ H m (f(A i )) ≤ λ m H m · i (A i ), and λ −1  · i ≤ |Df x (·)| ≤ λ ·  i for all x ∈ A i . This last assertion yields λ −m J m ( ·  i ) ≤ J m (Df x ) ≤ λ m J m ( ·  i ). We conclude that H m (f(A i )) ≤ λ m H m · i (A i ) = λ m J m ( ·  i )L m (A i ) ≤ λ 2m  A i J m (Df x ) dx. Since each f|A i is injective, summation over i gives  R n #(f −1 {y} ∩A) dH m (y) ≤ λ 2m  A J m (Df x ) dx. 9 Similarly, λ −2m  A J m (Df x ) dx ≤  R n #(f −1 {y} ∩A) dH m (y). As this holds for all λ > 1, the two integrals are equal. Case 2: A ⊂ {x: Df x exists and has rank < m}. Then J m (Df x ) = 0 for all x ∈ A. For  > 0, consider the map F : R m → R n × R m , F (x) = (f(x), x). For x ∈ A, it follows that DF x  ≤ Lip(f) +  and J m (DF x ) ≤ (Lip(f) + ) m−1 . Applying the result of the first case to F , we get H m (F (A)) =  A J m (DF x ) dx ≤ (Lip(f) + ) m−1 L m (A). Since H m (f(A)) ≤ H m (F (A)) for all  > 0, it follows that H m (f(A)) = 0. Thus both integrals equal 0. Case 3: A ⊂ {x : Df x does not exist}. Then H m (f(A)) ≤ Lip(f) m H m (A) = Lip(f) m L m (A) = 0 by Rademacher’s theorem. Thus both integrals equal 0. (2) follows from (1) by approximation. ✷ Rectifiable sets The following notion is fundamental in geometric measure theory. 1.13 Definition (countably rectifiable set) Let Y be a metric space. A set E ⊂ Y is countably H m -rectifiable if there is a sequence of Lipschitz maps f i : A i → Y , A i ⊂ R m , such that H m  E \  i f i (A i )  = 0. It is often possible to take w.l.o.g. A i = R m , e.g. if Y is a Banach space (recall Theorem 1.3). 1.14 Theorem (countably rectifiable sets in R n ) A set E ⊂ R n is countably H m -rectifiable if and only if there exists a se- quence of m-dimensional C 1 submanifolds M k of R n such that H m  E \  k M k  = 0. See [Fed, 3.2.29], [Sim, 11.1]. 10 [...]... 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Introduction to Geometric Measure Theory Urs Lang April 20, 2005 Abstract These are the notes to four one-hour lectures I delivered at the spring school Geometric Measure Theory: Old. follows from (1) by approximation. ✷ Rectifiable sets The following notion is fundamental in geometric measure theory. 1.13 Definition (countably rectifiable set) Let Y be a metric space. A set E ⊂ Y. U i : both a i and b i are differentiable at x}. By Rademacher’s theorem, Z :=  ∞ i=1 U i A i has measure zero. Let x ∈ L(f) Z. We show that for some i, x ∈ A i and a i (x) = b i (x); then f is