An introduction to Geometric Measure Theory Part 2: Hausdorff measure

Transcription

1 An introduction to Geometric Measure Theory Part 2: Hausdorff measure Toby O Neil, 10 October 2016 TCON (Open University) An introduction to GMT, part 2 10 October / 40

2 Last week... Discussed several motivating examples Defined box dimension Introduced Hausdorff measure and dimension TCON (Open University) An introduction to GMT, part 2 10 October / 40

3 Today 1 Determine the Hausdorff dimension of a simple set 2 Look at properties of Hausdorff measure. (Prove that it s a measure... ) 3 Find some useful alternative characterisations of Hausdorff dimension 4 Discuss an application TCON (Open University) An introduction to GMT, part 2 10 October / 40

4 Recall: covers Definition (r-covers) Let r > 0. A countable collection of sets {U i : i 2 N} in R n is an r-cover of E R n if 1 E S i2n U i 2 diam(u i ) apple r for each i 2 N. TCON (Open University) An introduction to GMT, part 2 10 October / 40

5 notation For a set A and s 0, we define 8 >< 0 if A= ;, A s = 1 if A 6= ; and s = 0, >: diam(a) s otherwise. TCON (Open University) An introduction to GMT, part 2 10 October / 40

6 Hausdorff measures Definition (s-dimensional Hausdorff measure) Suppose that F is a subset of R n and s 0. For any r > 0, we define ( 1 ) X Hr s (F) =inf U i s : {U i } is an r-cover of F. i=1 The s-dimensional Hausdorff measure is then given by H s (F) =lim r&0 H s r (F). TCON (Open University) An introduction to GMT, part 2 10 October / 40

7 Easy properties of Hausdorff measure 1 If s < t and H s (F) < 1, then H t (F) =0. (So for t > n, H t (R n )=0.) 2 If s is a non-negative integer, then H s is a constant multiple of the usual s-dimensional volume. (counting measure, length measure, area, volume etc.) we shall not prove this. TCON (Open University) An introduction to GMT, part 2 10 October / 40

8 More easy properties of Hausdorff measure 1 If 0 < r < inf{d(x, y) :x 2 E, y 2 F}, then H s r (E [ F) =H s r (E)+H s r (F) and so, if inf{d(x, y) :x 2 E, y 2 F} > 0, then H s (E [ F) =H s (E)+H s (F). 2 Let F R n and suppose that f : F! R m is such that for some fixed constants c and f (x) f (y) applec x y whenever x, y 2 F. Then for each s, H s/ (f (F)) apple c s/ H s (F). TCON (Open University) An introduction to GMT, part 2 10 October / 40

9 Hausdorff dimension Definition (Hausdorff dimension) For a set F, we define the Hausdorff dimension of F by dim H (F) =inf{s 0 : H s (F) =0} = sup{s : H s (F) =1}. Observations 1 Hausdorff dimension is defined for any set F (unlike box dimension). 2 dim H (;) =0 3 If A is a countable set, then dim H (A) =0. In particular, Q has Hausdorff dimension 0. TCON (Open University) An introduction to GMT, part 2 10 October / 40

10 Properties of Hausdorff dimension 1 If A B, then dim H (A) apple dim H (B). 2 If F 1, F 2, F 3,...is a (countable) sequence of sets, then 1! [ dim H F i = sup{dim H (F i ):1applei apple 1}. i=1 3 for bounded sets F, dim H (F) apple dim B (F). TCON (Open University) An introduction to GMT, part 2 10 October / 40

11 Calculating Hausdorff dimension ( )-Cantor set TCON (Open University) An introduction to GMT, part 2 10 October / 40

12 Heuristic for finding what the dimension could be Assume that F has positive and finite s-dimensional Hausdorff measure when s = dim H (F) and then represent F as a finite disjoint union of scaled copies of F, F i, say where F i is a copy of F scaled by i. Then! [ H s (F) =H s F i = X H s (F i )= X s i H s (F ). i i i Dividing through by H s (F) then gives 1 = X i s i. For ( 1 1 )-Cantor set obtain TCON (Open University) An introduction to GMT, part 2 10 October / 40

13 Covers provide an upper bound If set is bounded, then can use box dimension to find an upper bound, since in this case dim H (F) apple dim B (F). TCON (Open University) An introduction to GMT, part 2 10 October / 40

14 Upper bound for our example. TCON (Open University) An introduction to GMT, part 2 10 October / 40

15 TCON (Open University) An introduction to GMT, part 2 10 October / 40

16 Lower bound for our example TCON (Open University) An introduction to GMT, part 2 10 October / 40

17 TCON (Open University) An introduction to GMT, part 2 10 October / 40

18 Digression: some measure theory (X, d), a metric space. (Usually complete and separable) Definition: (outer) measures A set function µ: {A : A X}![0, 1] is a measure if 1 µ(;) =0 2 if A B, then µ(a) apple µ(b) 3 µ ( S 1 i=1 A i) apple P 1 i=1 µ(a i) Definition: measurable set A X is measurable if µ(e) =µ(e \ A)+µ(E \ A) for each E X TCON (Open University) An introduction to GMT, part 2 10 October / 40

19 Basic results Theorem If µ is a measure and M is the µ-measurable sets, then 1 M is a -algebra. [; 2M, closed under complements and countable unions] 2 if µ(a) =0, then A 2M 3 if A 1, A 2,...2Mare pairwise disjoint, then µ ( S A i )= P 1 i=1 µ(a i) 4 if A 1, A 2,...2M, then 1 µ ( S A i )=lim i!1 µ(a i ), provided A 1 A 2 2 µ ( T A i )=lim i!1 µ(a i ), provided A 1 A 2 and µ(a 1 ) < 1 TCON (Open University) An introduction to GMT, part 2 10 October / 40

20 Regular measures µ is a regular measure if for each A X, there is a µ-measurable set B with A B and µ(a) =µ(b). Lemma Suppose that µ is a regular measure. If A 1 A 2, then µ 1[ i=1 A i! = lim i!1 µ(a i ). TCON (Open University) An introduction to GMT, part 2 10 October / 40

21 Identifying measurable sets Definition The family of Borel sets in a metric space X is the smallest -algebra that contains the open subsets of X. Definition A measure µ is: 1 a Borel measure if the Borel sets are µ-measurable 2 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). TCON (Open University) An introduction to GMT, part 2 10 October / 40

22 Theorem Let µ be a measure on X. Then µ is a Borel measure if, and only if, µ(a [ B) =µ(a)+µ(b), whenever inf{d(x, y) :x 2 A, y 2 B} > 0. Theorem H s is a Borel regular measure for each s 0. TCON (Open University) An introduction to GMT, part 2 10 October / 40

23 Definition (support) The support of a Borel measure µ is defined to be the smallest closed set F for which µ(x \ F) =0. spt(µ) =X \ S {U : U is open and µ(u) =0}. Definition (Mass distribution) A Borel measure µ is a mass distribution on the set F if the support of µ is contained in F and 0 <µ(f) < 1. TCON (Open University) An introduction to GMT, part 2 10 October / 40

24 Mass distribution principle Suppose that µ is a mass distribution on a set F and for some real number s, there are c > 0 and r 0 > 0 so that µ(u) apple c U s if diam(u) apple r 0. Then H s (F) µ(f)/c > 0 and so dim H (F) s. TCON (Open University) An introduction to GMT, part 2 10 October / 40

25 Cantor set revisited TCON (Open University) An introduction to GMT, part 2 10 October / 40

26 TCON (Open University) An introduction to GMT, part 2 10 October / 40

27 Approximating sets Theorem Let µ be a Borel regular measure on X, let A be a µ-measurable set and let >0. 1 If µ(a) < 1, then there is a closed set C A for which µ(a \ C) <. 2 If there are open sets V 1, V 2,...with A S 1 i=1 and µ(v i ) < 1 for each i, then there is an open set V with A V and µ(v \ A) <. TCON (Open University) An introduction to GMT, part 2 10 October / 40

28 Radon measures Definition A Borel measure µ is a Radon measure on X if 1 all compact subsets of X have finite µ-measure 2 for open sets V, 3 for each set A X, µ(v )=sup{µ(k ):K V is compact} Corollary µ(a) =inf{µ(v ):V is open and A V }. A measure µ on R n is a Radon measure if and only if it is locally finite and Borel regular. TCON (Open University) An introduction to GMT, part 2 10 October / 40

29 Energies Definition For A R n, let M(A) denote the set of all compactly supported Radon measures µ with 0 <µ(a) < 1 and with support contained in A. Definition (t-energy) For a Radon measure µ on R n and t 0, we define the t-energy of µ by ZZ 1 I t (µ) = x y dµ(x)dµ(y). t (This may be infinite.) TCON (Open University) An introduction to GMT, part 2 10 October / 40

30 Aside: integration Theorem (Fubini) Let X and Y be separable metric spaces with µ and locally finite Borel measures on X and Y, respectively. If f is a non-negative Borel function on X Y, then ZZ ZZ f (x, y) dµ(x)d (y) = f (x, y) d (y)dµ(x). In particular, if f is the characteristic function of a Borel set, then Z Z µ({x :(x, y) 2 A}) d (y) = ({y :(x, y) 2 A}) dµ(x). TCON (Open University) An introduction to GMT, part 2 10 October / 40

31 More integration: a useful equation Theorem Let µ be a Borel measure and f a non-negative Borel function on a separable metric space X. Then Z Z 1 fdµ = µ({x 2 X : f (x) t}) dt. 0 TCON (Open University) An introduction to GMT, part 2 10 October / 40

32 Another view of I t (µ) I t (µ) =t Z 1 r t 0 1 µ(b(x, r)) dr TCON (Open University) An introduction to GMT, part 2 10 October / 40

33 A useful characterisation of dim H Theorem If A is a Borel set in R n, then dim H (A) =sup{t : There is µ 2Awith I t (µ) < 1}. (Finer results are available... ) TCON (Open University) An introduction to GMT, part 2 10 October / 40