Measuring sets with translation invariant Borel measures

Transcription

1 Measuring sets with translation invariant Borel measures University of Warwick Fractals and Related Fields III, 21 September 2015

2 Measured sets Definition A Borel set A R is called measured if there is a translation invariant Borel measure µ on R which assigns positive and σ-finite measure to A. sets with translation invariant measures 2 / 18

3 Measured sets Definition A Borel set A R is called measured if there is a translation invariant Borel measure µ on R which assigns positive and σ-finite measure to A. Question Is there a non-empty set which is not measured? sets with translation invariant measures 2 / 18

4 Measured sets Definition A Borel set A R is called measured if there is a translation invariant Borel measure µ on R which assigns positive and σ-finite measure to A. Question Is there a non-empty set which is not measured? translation invariant: µ(b + x) = µ(b) for every x R and Borel set B; µ(a) > 0 and µ restricted to A is σ-finite; µ on R need not be σ-finite. sets with translation invariant measures 2 / 18

5 Measured sets Definition A Borel set A R is called measured if there is a translation invariant Borel measure µ on R which assigns positive and σ-finite measure to A. Question Is there a non-empty set which is not measured? translation invariant: µ(b + x) = µ(b) for every x R and Borel set B; µ(a) > 0 and µ restricted to A is σ-finite; µ on R need not be σ-finite. Examples Any set of positive Lebesgue measure is measured by the Lebesgue measure; Cantor set is measured by the Hausdorff measure of dimension log 2/ log 3. sets with translation invariant measures 2 / 18

6 Measured sets Definition A Borel set A R is called measured if there is a translation invariant Borel measure µ on R which assigns positive and σ-finite measure to A. Question Is there a non-empty set which is not measured? translation invariant: µ(b + x) = µ(b) for every x R and Borel set B; µ(a) > 0 and µ restricted to A is σ-finite; µ on R need not be σ-finite. Examples Any set of positive Lebesgue measure is measured by the Lebesgue measure; Cantor set is measured by the Hausdorff measure of dimension log 2/ log 3. Nice examples of translation invariant measures: Hausdorff measures H s, generalised Hausdorff H g and packing measures P g with gauge function g. sets with translation invariant measures 2 / 18

7 Sets which have zero or infinite measure for every translation invariant Borel measure sets with translation invariant measures 3 / 18

8 Sets which have zero or infinite measure for every translation invariant Borel measure any infinite countable set; sets with translation invariant measures 3 / 18

9 Sets which have zero or infinite measure for every translation invariant Borel measure any infinite countable set; R. sets with translation invariant measures 3 / 18

10 A set which is not measured Theorem (Davies 1971) There is a non-empty compact set which is not measured. sets with translation invariant measures 4 / 18

11 A set which is not measured Theorem (Davies 1971) There is a non-empty compact set which is not measured. Sketch of proof. A = B (B + B) (B + B + B) (B + B + B + B)... sets with translation invariant measures 4 / 18

12 A set which is not measured Theorem (Davies 1971) There is a non-empty compact set which is not measured. Sketch of proof. A = B (B + B) (B + B + B) (B + B + B + B)... Let B [1, 2] be an uncountable compact set for which these unions are all disjoint. sets with translation invariant measures 4 / 18

13 A set which is not measured Theorem (Davies 1971) There is a non-empty compact set which is not measured. Sketch of proof. A = B (B + B) (B + B + B) (B + B + B + B)... Let B [1, 2] be an uncountable compact set for which these unions are all disjoint. B + B = b B (B + b). these as well sets with translation invariant measures 4 / 18

14 A set which is not measured Theorem (Davies 1971) There is a non-empty compact set which is not measured. Sketch of proof. A = B (B + B) (B + B + B) (B + B + B + B)... Let B [1, 2] be an uncountable compact set for which these unions are all disjoint. B + B = b B (B + b). these as well Lemma: If µ is translation invariant, µ(e) > 0 and F contains uncountably many disjoint translates of E, then µ is not σ-finite on F. sets with translation invariant measures 4 / 18

15 Other examples Theorem (Elekes Keleti 2006) The set of Liouville numbers L = {x R \ Q : for every n there are p, q with x p/q < 1/q n } is not measured. sets with translation invariant measures 5 / 18

16 Other examples Theorem (Elekes Keleti 2006) The set of Liouville numbers L = {x R \ Q : for every n there are p, q with x p/q < 1/q n } is not measured. Any non-f σ additive subgroup of R is not measured. sets with translation invariant measures 5 / 18

17 Other examples Theorem (Elekes Keleti 2006) The set of Liouville numbers L = {x R \ Q : for every n there are p, q with x p/q < 1/q n } is not measured. Any non-f σ additive subgroup of R is not measured. Remark σ-finite Borel measures on a Borel set A R are inner regular. sets with translation invariant measures 5 / 18

18 Theorem/Observation (Elekes Keleti) Let ν be a measure on A. Then ν has a translation invariant extension to R if and only if ν(a + t) = ν(a ) whenever A A and A + t A sets with translation invariant measures 6 / 18

19 Theorem/Observation (Elekes Keleti) Let ν be a measure on A. Then ν has a translation invariant extension to R if and only if ν(a + t) = ν(a ) whenever A A and A + t A Proof. { } µ(b) = inf ν(a i ) : A i A, B i (A i + t i ). i=1 sets with translation invariant measures 6 / 18

20 Theorem/Observation (Elekes Keleti) Let ν be a measure on A. Then ν has a translation invariant extension to R if and only if ν(a + t) = ν(a ) whenever A A and A + t A Proof. { } µ(b) = inf ν(a i ) : A i A, B i (A i + t i ). i=1 Remark If A is such that A (A + t) is at most 1 point for all t R, then any non-atomic measure on A is extendable. sets with translation invariant measures 6 / 18

21 Theorem/Observation (Elekes Keleti) Let ν be a measure on A. Then ν has a translation invariant extension to R if and only if ν(a + t) = ν(a ) whenever A A and A + t A Proof. { } µ(b) = inf ν(a i ) : A i A, B i (A i + t i ). i=1 Remark If A is such that A (A + t) is at most 1 point for all t R, then any non-atomic measure on A is extendable. Corollary If A is an uncountable Borel set such that A (A + t) is at most 1 point for all t R, then A is measured. sets with translation invariant measures 6 / 18

22 Questions (Elekes Keleti) 1 Is it true that the union of two measured sets is measured? sets with translation invariant measures 7 / 18

23 Questions (Elekes Keleti) 1 Is it true that the union of two measured sets is measured? 2 Is it true that every non-empty Borel set is a union of countably many measured sets? sets with translation invariant measures 7 / 18

24 Questions (Elekes Keleti) 1 Is it true that the union of two measured sets is measured? 2 Is it true that every non-empty Borel set is a union of countably many measured sets? 3 Is it true that every measured set is measured by a Hausdorff measure? sets with translation invariant measures 7 / 18

25 Questions (Elekes Keleti) 1 Is it true that the union of two measured sets is measured? 2 Is it true that every non-empty Borel set is a union of countably many measured sets? 3 Is it true that every measured set is measured by a Hausdorff measure? First aim: Find a set which is not measured but can be divided into two measured sets. sets with translation invariant measures 7 / 18

26 Union of measured sets in R The Hausdorff measure with gauge function g is { } H g (A) = lim inf g(diam U i ) : A i=1u i and diam U i < δ δ 0+ i=1 Here g is monotone increasing right continuous function g : [0, ) [0, ). sets with translation invariant measures 8 / 18

27 Union of measured sets in R The Hausdorff measure with gauge function g is { } H g (A) = lim inf g(diam U i ) : A i=1u i and diam U i < δ δ 0+ i=1 Here g is monotone increasing right continuous function g : [0, ) [0, ). Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. sets with translation invariant measures 8 / 18

28 Union of measured sets in R The Hausdorff measure with gauge function g is { } H g (A) = lim inf g(diam U i ) : A i=1u i and diam U i < δ δ 0+ i=1 Here g is monotone increasing right continuous function g : [0, ) [0, ). Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy sets with translation invariant measures 8 / 18

29 Union of measured sets in R The Hausdorff measure with gauge function g is { } H g (A) = lim inf g(diam U i ) : A i=1u i and diam U i < δ δ 0+ i=1 Here g is monotone increasing right continuous function g : [0, ) [0, ). Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy H g1 (B 1 ) = 1, H g1 (A 1 ) = 0, H g2 (B 2 ) = 1, H g2 (A 2 ) = 0. sets with translation invariant measures 8 / 18

30 Union of measured sets in R The Hausdorff measure with gauge function g is { } H g (A) = lim inf g(diam U i ) : A i=1u i and diam U i < δ δ 0+ i=1 Here g is monotone increasing right continuous function g : [0, ) [0, ). Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy H g1 (B 1 ) = 1, H g1 (A 1 ) = 0, H g2 (B 2 ) = 1, H g2 (A 2 ) = 0. The theorem holds even if A = B; or when B has Hausdorff dimension zero and A has Hausdorff dimension 1. sets with translation invariant measures 8 / 18

31 Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. sets with translation invariant measures 9 / 18

32 Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy sets with translation invariant measures 9 / 18

33 Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy H g1 (B 1 ) = 1, H g1 (A 1 ) = 0, H g2 (B 2 ) = 1, H g2 (A 2 ) = 0. sets with translation invariant measures 9 / 18

34 Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy H g1 (B 1 ) = 1, H g1 (A 1 ) = 0, H g2 (B 2 ) = 1, H g2 (A 2 ) = 0. Corollary Every Lebesgue nullset of the second category is a union of two measured sets. sets with translation invariant measures 9 / 18

35 Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy H g1 (B 1 ) = 1, H g1 (A 1 ) = 0, H g2 (B 2 ) = 1, H g2 (A 2 ) = 0. Corollary Every Lebesgue nullset of the second category is a union of two measured sets. There are Lebesgue nullsets of the second category which are not measured (e.g. set of Liouville numbers). sets with translation invariant measures 9 / 18

36 Theorem (M) Let A, B R be Borel sets of zero Lebesgue measure and assume that B is of the second category. Then there are Borel partitions B = B 1 B 2, A = A 1 A 2 and gauge functions g 1, g 2 such that the Hausdorff measures satisfy H g1 (B 1 ) = 1, H g1 (A 1 ) = 0, H g2 (B 2 ) = 1, H g2 (A 2 ) = 0. Corollary Every Lebesgue nullset of the second category is a union of two measured sets. There are Lebesgue nullsets of the second category which are not measured (e.g. set of Liouville numbers). Corollary The union of two measured sets need not be measured. sets with translation invariant measures 9 / 18

37 How to divide a set into two sets with translation invariant measures 10 / 18

38 How to divide a set into two Lemma (M) Let A R be a Borel set. Let g 1, g 2 be two gauge functions such that H min(g1,g2) (A) = 0. Then there are disjoint Borel sets A 1, A 2 such that A = A 1 A 2 and H g1 (A 1 ) = H g2 (A 2 ) = 0. sets with translation invariant measures 10 / 18

39 How to divide a set into two Lemma (M) Let A R be a Borel set. Let g 1, g 2 be two gauge functions such that H min(g1,g2) (A) = 0. Then there are disjoint Borel sets A 1, A 2 such that A = A 1 A 2 and H g1 (A 1 ) = H g2 (A 2 ) = 0. Proof. g(x) = min(g 1 (x), g 2 (x)). Cover A with intervals I 1, I 2,... such that j=1 g(i j) < ε. sets with translation invariant measures 10 / 18

40 How to divide a set into two Lemma (M) Let A R be a Borel set. Let g 1, g 2 be two gauge functions such that H min(g1,g2) (A) = 0. Then there are disjoint Borel sets A 1, A 2 such that A = A 1 A 2 and H g1 (A 1 ) = H g2 (A 2 ) = 0. Proof. g(x) = min(g 1 (x), g 2 (x)). Cover A with intervals I 1, I 2,... such that j=1 g(i j) < ε. Let S {1, 2,...} be the set of those j for which g(i j ) = g 1 (I j ). sets with translation invariant measures 10 / 18

41 How to divide a set into two Lemma (M) Let A R be a Borel set. Let g 1, g 2 be two gauge functions such that H min(g1,g2) (A) = 0. Then there are disjoint Borel sets A 1, A 2 such that A = A 1 A 2 and H g1 (A 1 ) = H g2 (A 2 ) = 0. Proof. g(x) = min(g 1 (x), g 2 (x)). Cover A with intervals I 1, I 2,... such that j=1 g(i j) < ε. Let S {1, 2,...} be the set of those j for which g(i j ) = g 1 (I j ). Then g 1 (I j ) + g 2 (I j ) < ε. j S j S sets with translation invariant measures 10 / 18

42 How to divide a set into two Lemma (M) Let A R be a Borel set. Let g 1, g 2 be two gauge functions such that H min(g1,g2) (A) = 0. Then there are disjoint Borel sets A 1, A 2 such that A = A 1 A 2 and H g1 (A 1 ) = H g2 (A 2 ) = 0. Proof. g(x) = min(g 1 (x), g 2 (x)). Cover A with intervals I 1, I 2,... such that j=1 g(i j) < ε. Let S {1, 2,...} be the set of those j for which g(i j ) = g 1 (I j ). Then g 1 (I j ) + g 2 (I j ) < ε. j S j S A 1 j S I j A 2 j S I j (use limsup and liminf sets) sets with translation invariant measures 10 / 18

43 Technical part of the proof Proposition (M) Let B R be a Borel set of the second Baire category and let A R have Lebesgue measure zero. Then there are gauge functions g 1, g 2 such that H g1 (B) > 0, H g2 (B) > 0, and H min(g1,g2) (A) = 0. sets with translation invariant measures 11 / 18

44 Technical part of the proof Proposition (M) Let B R be a Borel set of the second Baire category and let A R have Lebesgue measure zero. Then there are gauge functions g 1, g 2 such that H g1 (B) > 0, H g2 (B) > 0, and H min(g1,g2) (A) = 0. About the proof: sets with translation invariant measures 11 / 18

45 Technical part of the proof Proposition (M) Let B R be a Borel set of the second Baire category and let A R have Lebesgue measure zero. Then there are gauge functions g 1, g 2 such that H g1 (B) > 0, H g2 (B) > 0, and H min(g1,g2) (A) = 0. About the proof: 1 It is very easy to construct compact sets in dense G δ sets. sets with translation invariant measures 11 / 18

46 Technical part of the proof Proposition (M) Let B R be a Borel set of the second Baire category and let A R have Lebesgue measure zero. Then there are gauge functions g 1, g 2 such that H g1 (B) > 0, H g2 (B) > 0, and H min(g1,g2) (A) = 0. About the proof: 1 It is very easy to construct compact sets in dense G δ sets. 2 Two compact sets K i and gauge functions g i are constructed such that H gi (K i ) = 1. sets with translation invariant measures 11 / 18

47 Technical part of the proof Proposition (M) Let B R be a Borel set of the second Baire category and let A R have Lebesgue measure zero. Then there are gauge functions g 1, g 2 such that H g1 (B) > 0, H g2 (B) > 0, and H min(g1,g2) (A) = 0. About the proof: 1 It is very easy to construct compact sets in dense G δ sets. 2 Two compact sets K i and gauge functions g i are constructed such that H gi (K i ) = 1. 3 min(g 1, g 2 ) will be linear on long intervals, [ε, δ], ε δ. sets with translation invariant measures 11 / 18

48 Technical part of the proof Proposition (M) Let B R be a Borel set of the second Baire category and let A R have Lebesgue measure zero. Then there are gauge functions g 1, g 2 such that H g1 (B) > 0, H g2 (B) > 0, and H min(g1,g2) (A) = 0. About the proof: 1 It is very easy to construct compact sets in dense G δ sets. 2 Two compact sets K i and gauge functions g i are constructed such that H gi (K i ) = 1. 3 min(g 1, g 2 ) will be linear on long intervals, [ε, δ], ε δ. 4 Linear gauge functions define constant times Lebesgue measure, so we will have H min(g1,g2) (A) = 0. sets with translation invariant measures 11 / 18

49 Technical part of the proof Proposition (M) Let B R be a Borel set of the second Baire category and let A R have Lebesgue measure zero. Then there are gauge functions g 1, g 2 such that H g1 (B) > 0, H g2 (B) > 0, and H min(g1,g2) (A) = 0. About the proof: 1 It is very easy to construct compact sets in dense G δ sets. 2 Two compact sets K i and gauge functions g i are constructed such that H gi (K i ) = 1. 3 min(g 1, g 2 ) will be linear on long intervals, [ε, δ], ε δ. 4 Linear gauge functions define constant times Lebesgue measure, so we will have H min(g1,g2) (A) = 0. Lemma + Proposition Theorem sets with translation invariant measures 11 / 18

50 Polish groups G is a Polish group if it is topological group homeomorphic to a separable complete metric space. sets with translation invariant measures 12 / 18

51 Polish groups G is a Polish group if it is topological group homeomorphic to a separable complete metric space. Example: Banach spaces; closed subgroups of Banach spaces. sets with translation invariant measures 12 / 18

52 Polish groups G is a Polish group if it is topological group homeomorphic to a separable complete metric space. Example: Banach spaces; closed subgroups of Banach spaces. Definition A Borel set A G is measured if there is a (both left and right) translation invariant Borel measure µ on G which assigns positive and σ-finite measure to A. sets with translation invariant measures 12 / 18

53 Polish groups G is a Polish group if it is topological group homeomorphic to a separable complete metric space. Example: Banach spaces; closed subgroups of Banach spaces. Definition A Borel set A G is measured if there is a (both left and right) translation invariant Borel measure µ on G which assigns positive and σ-finite measure to A. Theorem (M) A non-locally compact Polish group G is not a union of countably many measured sets. sets with translation invariant measures 12 / 18

54 Polish groups G is a Polish group if it is topological group homeomorphic to a separable complete metric space. Example: Banach spaces; closed subgroups of Banach spaces. Definition A Borel set A G is measured if there is a (both left and right) translation invariant Borel measure µ on G which assigns positive and σ-finite measure to A. Theorem (M) A non-locally compact Polish group G is not a union of countably many measured sets. Beginning of the proof. Assume it is. G = i A i. µ i. small diameter such that µ i (K i ) > 0. Choose compact sets K i A i of sets with translation invariant measures 12 / 18

55 Polish groups G is a Polish group if it is topological group homeomorphic to a separable complete metric space. Example: Banach spaces; closed subgroups of Banach spaces. Definition A Borel set A G is measured if there is a (both left and right) translation invariant Borel measure µ on G which assigns positive and σ-finite measure to A. Theorem (M) A non-locally compact Polish group G is not a union of countably many measured sets. Beginning of the proof. Assume it is. G = i A i. µ i. small diameter such that µ i (K i ) > 0. Take K = K 1 + K 2 + K , Choose compact sets K i A i of sets with translation invariant measures 12 / 18

56 Polish groups G is a Polish group if it is topological group homeomorphic to a separable complete metric space. Example: Banach spaces; closed subgroups of Banach spaces. Definition A Borel set A G is measured if there is a (both left and right) translation invariant Borel measure µ on G which assigns positive and σ-finite measure to A. Theorem (M) A non-locally compact Polish group G is not a union of countably many measured sets. Beginning of the proof. Assume it is. G = i A i. µ i. Choose compact sets K i A i of small diameter such that µ i (K i ) > 0. Take K = K 1 + K 2 + K , and consider the infinite convolution of the normalised measures µ i Ki µ i (K i ).... sets with translation invariant measures 12 / 18

57 Haar null (shy) sets On locally compact groups we have Haar measures. sets with translation invariant measures 13 / 18

58 Haar null (shy) sets On locally compact groups we have Haar measures. On non-locally compact groups there is no Haar measure, but there is a σ-ideal of Haar null sets. Definition (Christensen) A Borel set A G is called Haar null if there is a Borel probability measure µ such that µ(g + A + h) = 0 for all g, h G. sets with translation invariant measures 13 / 18

59 Haar null (shy) sets On locally compact groups we have Haar measures. On non-locally compact groups there is no Haar measure, but there is a σ-ideal of Haar null sets. Definition (Christensen) A Borel set A G is called Haar null if there is a Borel probability measure µ such that µ(g + A + h) = 0 for all g, h G. If G is locally compact, Haar null sets are the sets of zero Haar measure. sets with translation invariant measures 13 / 18

60 Haar null (shy) sets On locally compact groups we have Haar measures. On non-locally compact groups there is no Haar measure, but there is a σ-ideal of Haar null sets. Definition (Christensen) A Borel set A G is called Haar null if there is a Borel probability measure µ such that µ(g + A + h) = 0 for all g, h G. If G is locally compact, Haar null sets are the sets of zero Haar measure. Haar null sets are closed under countable unions. sets with translation invariant measures 13 / 18

61 Haar null (shy) sets On locally compact groups we have Haar measures. On non-locally compact groups there is no Haar measure, but there is a σ-ideal of Haar null sets. Definition (Christensen) A Borel set A G is called Haar null if there is a Borel probability measure µ such that µ(g + A + h) = 0 for all g, h G. If G is locally compact, Haar null sets are the sets of zero Haar measure. Haar null sets are closed under countable unions. prevalent sets. sets with translation invariant measures 13 / 18

62 Haar null (shy) sets On locally compact groups we have Haar measures. On non-locally compact groups there is no Haar measure, but there is a σ-ideal of Haar null sets. Definition (Christensen) A Borel set A G is called Haar null if there is a Borel probability measure µ such that µ(g + A + h) = 0 for all g, h G. If G is locally compact, Haar null sets are the sets of zero Haar measure. Haar null sets are closed under countable unions. prevalent sets. The proof that Haar null sets form a non-trivial σ-ideal is essentially the same as the previous Beginning of the proof. sets with translation invariant measures 13 / 18

63 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. sets with translation invariant measures 14 / 18

64 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. sets with translation invariant measures 14 / 18

65 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. (Or let X be a Banach space with a normalised Schauder basis (e n ).) sets with translation invariant measures 14 / 18

66 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. (Or let X be a Banach space with a normalised Schauder basis (e n ).) Let G = { n=1 k n n e n X : k n Z } sets with translation invariant measures 14 / 18

67 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. (Or let X be a Banach space with a normalised Schauder basis (e n ).) Let G = { n=1 k n n e n X : k n Z } = ( Z 1 Z 2 Z ) 3... X. sets with translation invariant measures 14 / 18

68 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. (Or let X be a Banach space with a normalised Schauder basis (e n ).) Let G = { n=1 k n n e n X : k n Z This is the closure of the subgroup generated by (e n /n). } = ( Z 1 Z 2 Z ) 3... X. sets with translation invariant measures 14 / 18

69 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. (Or let X be a Banach space with a normalised Schauder basis (e n ).) Let G = { n=1 k n n e n X : k n Z This is the closure of the subgroup generated by (e n /n). Note: if the sum converges, then kn n } = ( Z 1 Z 2 Z ) 3... X. kn is bounded and n C k n n=1 n e n. sets with translation invariant measures 14 / 18

70 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. (Or let X be a Banach space with a normalised Schauder basis (e n ).) Let G = { n=1 k n n e n X : k n Z This is the closure of the subgroup generated by (e n /n). Note: if the sum converges, then kn n n=1 } = ( Z 1 Z 2 Z ) 3... X. kn is bounded and n C k n n=1 n e n. Define a projection f : G R by setting ( ) k n f n e k n n = n!. n=1 sets with translation invariant measures 14 / 18

71 Theorem (M) There is a non-empty compact set in R which is not a union of countably many measured sets. Consider a Banach space X = l p, 1 p <, and let (e n ) be the standard basis. (Or let X be a Banach space with a normalised Schauder basis (e n ).) Let G = { n=1 k n n e n X : k n Z This is the closure of the subgroup generated by (e n /n). Note: if the sum converges, then kn n n=1 } = ( Z 1 Z 2 Z ) 3... X. kn is bounded and n C k n n=1 n e n. Define a projection f : G R by setting ( ) k n f n e k n n = n!. f is a (restriction of a) bounded linear functional. n=1 sets with translation invariant measures 14 / 18

72 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f n=1 k n n e n ) = n=1 k n n!. sets with translation invariant measures 15 / 18

73 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f X r = {x X : x r}. n=1 k n n e n ) = n=1 k n n!. sets with translation invariant measures 15 / 18

74 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f X r = {x X : x r}. n=1 k n n e n ) = n=1 k n n!. f is a bounded linear functional, f (G) is an additive subgroup of R. sets with translation invariant measures 15 / 18

75 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f X r = {x X : x r}. n=1 k n n e n ) = n=1 k n n!. f is a bounded linear functional, f (G) is an additive subgroup of R. f is locally injective (injective on G X r for small enough r) sets with translation invariant measures 15 / 18

76 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f X r = {x X : x r}. n=1 k n n e n ) = n=1 k n n!. f is a bounded linear functional, f (G) is an additive subgroup of R. f is locally injective (injective on G X r for small enough r) f (G X r ) is Borel; thus f (G) R is Borel. sets with translation invariant measures 15 / 18

77 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f X r = {x X : x r}. n=1 k n n e n ) = n=1 k n n!. f is a bounded linear functional, f (G) is an additive subgroup of R. f is locally injective (injective on G X r for small enough r) f (G X r ) is Borel; thus f (G) R is Borel. G is a non-locally compact Polish space: not a union of countably many measured sets. sets with translation invariant measures 15 / 18

78 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f X r = {x X : x r}. n=1 k n n e n ) = n=1 k n n!. f is a bounded linear functional, f (G) is an additive subgroup of R. f is locally injective (injective on G X r for small enough r) f (G X r ) is Borel; thus f (G) R is Borel. G is a non-locally compact Polish space: not a union of countably many measured sets. By local injectivity, we can pull back measures on f (G) to measures on G. Therefore A = f (G) is not a union of countably many measured sets. sets with translation invariant measures 15 / 18

79 X = l p, 1 p <, with standard basis (e n ). { } ( k n Z G = n e n X : k n Z = 1 Z 2 Z ) 3... X. n=1 Define a projection f : G R by ( f X r = {x X : x r}. n=1 k n n e n ) = n=1 k n n!. f is a bounded linear functional, f (G) is an additive subgroup of R. f is locally injective (injective on G X r for small enough r) f (G X r ) is Borel; thus f (G) R is Borel. G is a non-locally compact Polish space: not a union of countably many measured sets. By local injectivity, we can pull back measures on f (G) to measures on G. Therefore A = f (G) is not a union of countably many measured sets. If (e n ) is boundedly complete, A = f (G) is σ-compact. sets with translation invariant measures 15 / 18

80 Typical compact sets Non-empty compact subsets of R with the Hausdorff distance form a complete separable metric space. sets with translation invariant measures 16 / 18

81 Typical compact sets Non-empty compact subsets of R with the Hausdorff distance form a complete separable metric space. A typical compact set in R is measured. sets with translation invariant measures 16 / 18

82 Typical compact sets Non-empty compact subsets of R with the Hausdorff distance form a complete separable metric space. A typical compact set in R is measured. A typical compact set K satisfies that for all t R, K (K + t) is at most 1 point. sets with translation invariant measures 16 / 18

83 Typical compact sets Non-empty compact subsets of R with the Hausdorff distance form a complete separable metric space. A typical compact set in R is measured. A typical compact set K satisfies that for all t R, K (K + t) is at most 1 point. All such sets are measured. sets with translation invariant measures 16 / 18

84 Typical compact sets Non-empty compact subsets of R with the Hausdorff distance form a complete separable metric space. A typical compact set in R is measured. A typical compact set K satisfies that for all t R, K (K + t) is at most 1 point. All such sets are measured. Theorem (Balka M) For a typical compact set K R there is a gauge function g with H g (K) = 1. sets with translation invariant measures 16 / 18

85 A measured set which is not measured by Hausdorff measures sets with translation invariant measures 17 / 18

86 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). sets with translation invariant measures 17 / 18

87 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. sets with translation invariant measures 17 / 18

88 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. Let A = P 1 P 2 P n n=1 sets with translation invariant measures 17 / 18

89 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. Let B = log A = A = P 1 P 2 P n n=1 (log P 1 + log P log P n ) n=1 sets with translation invariant measures 17 / 18

90 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. Let B = log A = A = P 1 P 2 P n n=1 (log P 1 + log P log P n ) n=1 Since P is algebraically independent, the equation x y = u v in A only has trivial solutions. Therefore A is measured. sets with translation invariant measures 17 / 18

91 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. Let B = log A = A = P 1 P 2 P n n=1 (log P 1 + log P log P n ) n=1 Since P is algebraically independent, the equation x y = u v in A only has trivial solutions. Therefore A is measured. B is not measured: this is basically Davies s example. sets with translation invariant measures 17 / 18

92 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. Let B = log A = A = P 1 P 2 P n n=1 (log P 1 + log P log P n ) n=1 Since P is algebraically independent, the equation x y = u v in A only has trivial solutions. Therefore A is measured. B is not measured: this is basically Davies s example. B is a bi-lipschitz image of A. sets with translation invariant measures 17 / 18

93 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. Let B = log A = A = P 1 P 2 P n n=1 (log P 1 + log P log P n ) n=1 Since P is algebraically independent, the equation x y = u v in A only has trivial solutions. Therefore A is measured. B is not measured: this is basically Davies s example. B is a bi-lipschitz image of A. If there was g such that H g (A) is positive and σ-finite, the same would hold for B. sets with translation invariant measures 17 / 18

94 A measured set which is not measured by Hausdorff measures John von Neumann: there is an algebraically independent non-empty perfect set P in (2, ). Find disjoint perfect sets P i (i = 1, 2,...) in P. Let B = log A = A = P 1 P 2 P n n=1 (log P 1 + log P log P n ) n=1 Since P is algebraically independent, the equation x y = u v in A only has trivial solutions. Therefore A is measured. B is not measured: this is basically Davies s example. B is a bi-lipschitz image of A. If there was g such that H g (A) is positive and σ-finite, the same would hold for B. So A is measured but not by Hausdorff measures. sets with translation invariant measures 17 / 18

95 A measured set which is not measured by packing measures Unfortunately 0 < P g (A) < and B is a bi-lipschitz image of A 0 < P g (B) < sets with translation invariant measures 18 / 18

96 A measured set which is not measured by packing measures Unfortunately 0 < P g (A) < and B is a bi-lipschitz image of A 0 < P g (B) < (Reference?) sets with translation invariant measures 18 / 18

97 A measured set which is not measured by packing measures Unfortunately 0 < P g (A) < and B is a bi-lipschitz image of A 0 < P g (B) < (Reference?) Fortunately Results of Peres and Elekes Keleti imply that some Befrod McMullen carpets are measured but not measured by packing measures. sets with translation invariant measures 18 / 18