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Notes on Rectifiability Notes on Rectifiability Urs Lang 2007 Version of September 1, 2016 Abstract These are the notes to a first part of a lecture course on Geometric Measure Theory I taught at ETH Zurich in Fall 2006. They partially overlap with [Lan1]. Contents 1 Embeddings of metric spaces 2 2 Compactness theorems for metric spaces 3 3 Lipschitz maps 5 4 Differentiability of Lipschitz maps 8 5 Extension of smooth functions 11 6 Metric differentiability 11 7 Hausdorff measures 15 8 Area formula 17 9 Rectifiable sets 20 10 Coarea formula 22 References 29 1 1 Embeddings of metric spaces We start with some basic and well-known isometric embedding theorems for metric spaces. 1 1 For a non-empty set X, l (X) = (l (X); k · k1) denotes the Banach space of all functions s: X ! R with ksk1 := supx2X js(x)j < 1. Similarly, 1 1 l := l (N) = f(sk)k2N : k(sk)k1 := supk2N jskj < 1g. 1.1 Proposition (Kuratowski, Fr´echet) (1) Every metric space X admits an isometric embedding into l1(X). (2) Every separable metric space admits an isometric embedding into l1. Proof : (1) Fix a basepoint z 2 X and define f : X ! l1(X), x 7! sx; sx(y) = d(x; y) − d(y; z): x x Note that ks k1 = supy js (y)j ≤ d(x; z). Moreover, x x0 0 0 ks − s k1 = supyjd(x; y) − d(x ; y)j ≤ d(x; x ); and equality occurs for y = x0. (2) Choose a basepoint z 2 X and a countable dense set D = fxk : k 2 1 Ng in X. Define f : X ! l , x 7! (d(x; xk) − d(xk; z))k2N: 1 1 By (1), fjD : D ! l = l (D) is an isometric embedding. Since D is dense and f is continuous, f is an isometric embedding. 2 Note that if X is bounded, we do not need to subtract the term d(y; z) or d(xk; z), respectively, in the definition of f. In this case, the embedding is canonical. For further reading on results of this type and detailed references we refer to [Hei2]. Recall that a metric space X is said to be precompact or totally bounded if for every > 0, X can be covered by a finite number of closed balls of radius . We call a set Y ⊂ X -separated if d(y; y0) ≥ whenever y; y0 2 Y , y 6= y0. Note that X is precompact if and only if for every > 0, all - separated subsets of X are finite. A metric space is compact if and only if it is precompact and complete. 1.2 Definition (uniformly precompact family) A family (Xj)j2J of metric spaces is called uniformly precompact if for all > 0 there exists a number n = n() 2 N such that each Xj can be covered by n closed balls of radius . The family (Xj)j2J is uniformly bounded if supj2J diam Xj < 1. 2 1.3 Theorem (Gromov embedding) Suppose that (Xj)j2J is a uniformly precompact and uniformly bounded family of metric spaces. Then there is a compact metric space Z such that each Xj admits an isometric embedding into Z. We follow essentially the original proof from [Gro1]. −i Proof : For i 2 N, let i := 2 and pick ni 2 N such that each Xj can be covered by ni closed balls of radius i. Choose a partition of N into sets Ni, i 2 N, with cardinality #Ni = n1n2 : : : ni, and define a map π : N n N1 ! N such that for each i 2 N, −1 −1 π (Ni) = Ni+1; #π fkg = ni+1 8k 2 Ni: j In each Xj, we construct a sequence (xk)k2N according to the following j inductive scheme. For i = 1, the points xk with k 2 Ni = N1 are chosen j such that the n1 balls B(xk; 1) cover Xj. For i ≥ 1, if the n1 : : : ni centers j j xk with k 2 Ni are selected, the n1 : : : nini+1 points xl with l 2 Ni+1 are j chosen such that for each k 2 Ni, the ball B(xk; i) is covered by the ni+1 balls j j B(xl ; i+1) ⊂ B(xk; 2i) with l 2 π−1fkg. This way we obtain for every j 2 J a dense sequence j 1 (xk)k2N in Xj which gives rise to an isometric embedding fj : Xj ! l , j −1 mapping x to (d(x; xk))k2N. Whenever i 2 N, k 2 Ni, and l 2 π fkg, then j j j j jd(x; xk) − d(x; xl )j ≤ d(xk; xl ) ≤ 2i: Hence, each fj(Xj) lies in the set Z of all sequences (sk)k2N with 0 ≤ sk ≤ supj diam Xj for all k 2 N and −1 jsk − slj ≤ 2i 8i 2 N; k 2 Ni; l 2 π fkg: Since the sequence (i)i2N is summable, it follows that Z is a compact subset of l1. 2 2 Compactness theorems for metric spaces For subsets A; B of a metric space X we denote by Nδ(A) = fx 2 X : d(x; A) ≤ δg the closed δ-neighborhood of A and by dH (A; B) = inffδ ≥ 0 : A ⊂ Nδ(B);B ⊂ Nδ(A)g the Hausdorff distance of A and B; dH defines a metric on the set C of non-empty, closed and bounded subsets of X. 3 2.1 Theorem (Blaschke) Suppose that X = (X; d) is a metric space and C is the set of non-empty, closed and bounded subsets of X, endowed with the Hausdorff metric dH . (1) If X is complete, then C is complete. (2) If X is compact, then C is compact. 3 This was first proved by Blaschke [Bla] for compact convex bodies in R to settle the existence question in the isoperimetric problem. Proof : (1): Let (Ci)i2N be a Cauchy sequence in C. Then the set T1 S C := i=1 j≥iCj is closed and bounded. We show that lim dH (Ci;C) = 0: i!1 Let > 0. Choose i0 such that dH (Ci;Cj) < /2 whenever i; j ≥ i0. Suppose x 2 C. Since C ⊂ S C , there exists an index j ≥ i with d(x; C ) < /2. j≥i0 j 0 j Hence d(x; Ci) ≤ d(x; Cj) + dH (Ci;Cj) < for all i ≥ i0. This shows that C ⊂ N(Ci) for i ≥ i0. Now suppose x 2 Ci for some i ≥ i0. Pick a sequence i = i1 < i2 < : : : k such that dH (Cm;Cn) < /2 whenever m; n ≥ ik, k 2 N. Then choose a k sequence (xk)k2N such that x1 = x, xk 2 Cik and d(xk; xk+1) < /2 . As X is complete, the Cauchy sequence (xk) converges to some point y. We have 1 X d(x; y) = lim d(x; xk) ≤ d(xk; xk+1) < , k!1 k=1 and y belongs to the closure of Cik [ Cik+1 [ ::: for all k. Thus y 2 C and d(x; C) < . This shows that Ci ⊂ N(C) whenever i ≥ i0. (2): We know that C is complete since X is, so it suffices to show that C is precompact. Let > 0. Since X is precompact, there exists a finite set Z ⊂ X with N(Z) = X. We show that every C 2 C is at Hausdorff distance at most of some subset of Z, namely ZC := Z \ N(C). For every x 2 C there exists a point z 2 Z with d(x; z) ≤ , so z 2 ZC . This shows that C ⊂ N(ZC ). Since also ZC ⊂ N(C), we have dH (C; ZC ) ≤ . As there are only finitely many distinct subsets of Z, we conclude that C is precompact. 2 2.2 Definition (Gromov–Hausdorff distance) The Gromov–Hausdorff distance of two metric spaces X; Y is the infimum of all r > 0 for which there exist a metric space (Z; dZ ) and subspaces X0 ⊂ Z 0 Z 0 0 and Y ⊂ Z isometric to X and Y , respectively, such that dH (X ;Y ) < r. 4 Cf. [Gro1], [Gro2]. Alternatively, call a metric d¯ on the disjoint union X t Y admissible for the given metrics d = dX and d = dY on X and Y if ¯ X ¯ Y djX×X = d and djY ×Y = d ; then ¯ dGH (X; Y ) = inf dH (X; Y ) where the infimum is taken over all admissible metrics d¯ on X t Y . For instance, suppose that diam(X); diam(Y ) ≤ D < 1. Setting d¯(x; y) = D=2 for x 2 X and y 2 Y we obtain an admissible metric on X t Y , in particular dGH (X; Y ) ≤ D=2. 2.3 Proposition (1) dGH satisfies the triangle inequality, i.e. dGH (X; Z) ≤ dGH (X; Y ) + dGH (Y; Z) for all metric spaces X; Y; Z. (2) dGH defines a metric on the set of isometry classes of compact metric spaces. See [BurBI, Proposition 7.3.16, Theorem 7.3.30]. Assertion (2) is no longer true if 'compact' is replaced by 'complete and bounded'. 2.4 Theorem (Gromov compactness criterion) Suppose that (Xi)i2N is a uniformly precompact and uniformly bounded sequence of metric spaces. Then there exist a subsequence (Xij )j2N and a compact metric space Z such that (Xij ) Gromov–Hausdorff converges to Z, i.e. limj!1 dGH (Xij ;Z) = 0. This was proved in [Gro1]. Proof : Combine Theorems 1.3 (Gromov embedding) and 2.1(2) (Blaschke). 2 3 Lipschitz maps Let X; Y be metric spaces, and let λ 2 [0; 1).
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