Introduction. Hausdorff Measure on O-Minimal Structures. Outline

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1 Introduction and Motivation Introduction Hausdorff Measure on O-Minimal Structures Integral geometry on o-minimal structures Antongiulio Fornasiero Università di Pisa Conference around o-minimality, Paris, September 2006 In [Berarducci-Otero, 2004], the authors define an analogue of Lebesgue measure for bounded definable subsets of an o-minimal structure, which expands a real closed field. Our aim is defining the area of a bounded definable d-dimensional subsets of n. Different possible definitions for such area are possible: we will prove that some of them are equivalent. On the reals, there are many different definitions of the d-area of a subset. However, on smooth sub-manifolds of R n all these definitions coincide with the d-dimensional Hausdorff measure H d. Every definable set in an o-minimal structure on the reals is a finite union of smooth sub-manifolds, therefore the d-area of such a set is H d. Many formulae are known for the Hausdorff measure: from Fubini s theorem to the area and coarea formulae. We will generalise some of these formulae to our measure on. Elisa Vasquez Rifo obtained independently some of the results exposed here. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / Berarducci-Otero measure is a real number, not an element of. The same holds for our measure. 2. Elisa is a student of Speissegger. 3. She works expecially in dimension 1 (curves). Outline 1 and Integral-Geometric Measure Measures and Outer Measures Hausdorff Measure on the Reals O-Minimal Lebesgue Measure Measure and Standard Part The Grassmannian and Integral-Geometric Measure Integral-Geometric Measure on O-Minimal Structures A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

2 Measures and Outer Measures Measures and Outer Measures Outer Measure The extended real line is R := R { ± }. Let Y be a set, Y be a family of subsets of Y, and µ : Y R. µ is monotone iff for every A, B Y, A B µ(a) µ(b). µ is countably subadditive (or σ-subadditive) iff for every A, B i Y A µ(b i ). i N B i µ(a) i N µ is countably additive (or σ-additive) iff for every B i Y B i disjoint and B i Y µ ( ) B i = µ(b i ). i N An outer measure on Y is a non-negative, monotone, and countably subadditive function µ : P(Y ) R, such that µ ( ) = 0. i N i N Measure Definition (Boolean rings and σ-algebrae) A Boolean ring (or simply ring) R on Y is a non-empty class of subsets of Y closed under finite union and set-difference. R is a σ-ring iff it is also closed under countable unions. R is a σ-algebra iff moreover Y R. Definition (Measure) A measure on a ring R is a map µ : R R which is monotone and countably additive, and such that µ( ) = 0. A measure is σ-finite iff Y is a countable union of sets in R of finite measure. A measure is complete iff for every B R and A B such that µ(b) = 0, we have A R (and consequently µ(a) = 0). A measure tout court is a measure on a σ-algebra on Y. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Measures and Outer Measures Hausdorff Measure on the Reals Carathéodory s Construction Hausdorff Measure Let µ be an outer measure on Y. A subset A Y is µ -measurable iff for every subset E Y, µ (E) = µ (E A) + µ (E \ A). The µ -measurable subsets of Y form a σ-algebra S µ. We will denote by µ the measure induced by µ, namely the restriction of µ to S µ. µ is a complete measure on S µ. n natural number; d real number, with 0 d n; A subset of R n ; H d (A) = H d,n (A) the d-dimensional Hausdorff measure of A. H d,n does not depends on n. It is an outer measure on R n. For every A R n there exists a unique d 0 R such that d > d 0 d < d 0 H d (A) = 0 and H d (A) = +. d 0 is the Hausdorff dimension of A. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

3 Hausdorff Measure on Manifolds Hausdorff Measure on the Reals Hausdorff Measure on the Reals Hausdorff Measure and Real O-Minimal Structures If A is an embedded C 1 manifold of dimension d, then d is equal to the Hausdorff dimension of A, and 0 < H d (A). If f : D R n is a C 1 map, with D R d, H d( f (D) ) is computed as the open integral of the Jacobian J d (f ) over D. The area of a d-dimensional manifold A R n is defined by calculating it on parametrised portions of A. Example d = 0, H 0 (A) = #(A) cardinality. d = 1, H 1 (A) length. d = 2, H 2 (A) area. d = n, H n,n (A) = L n (A) Lebesgue measure. R o-minimal structure on the real line R, expanding the ordered field structure. A R n definable (with parameters) in R. Then, The o-minimal dimension and the Hausdorff dimension of A coincide. 0 < H d (A), where d = dim(a). If A is bounded, then H d (A) < +. If A t is a uniformly bounded definable family of subsets of R n, then H d (A t ) is uniformly bounded. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Berarducci-Otero Definition O-Minimal Lebesgue Measure is an ℵ 1 -saturated o-minimal structure extending a real closed field. A subset A n is bounded iff there exists m N such that x < n for every x A. Fin() is the set of bounded elements. The family of bounded definable subsets of n is a boolean ring. Berarducci and Otero define an analogue of Lebesgue measure on Fin() n. More precisely, they define the outer measure L n on Fin() n as 1. L n is defined even when is not ℵ 1-saturated. However, when is countable, L n is not σ-additive, but only additive. L n (A) := inf ( k L n (Q i ) ), for every A Fin() n, where each Q i is a n-dimensional hypercube with rational vertices, such that k A Q i, and L n (Q i ) := l n i, where l i is the length of the edge of Q i. i=1 i=1 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

4 Outer Measure Pullback Measure and Standard Part Measure and Standard Part Standard Part and Baisalov-Poizat X, Y sets; ν an outer measure on Y ; f : X Y a surjective function. For every A X, define µ (A) := ν ( f (A) ) µ is an outer measure on X. A X is µ -measurable iff 1 f (A) is ν -measurable, and 2 µ ( Ã ) = 0. Ã := f 1( f (A) ) f 1( f (X \ A) ). ℵ 1 -saturated o-minimal structure expanding an ordered field. Fin() bounded elements of. equivalence relation: x y iff x y < 1 n for every n N. Fin()/ is naturally isomorphic to R. st : Fin() R quotient map = standard part map. R R induces a structure R on the reals via the map st. = expansion of R generated by the subsets of R n of the form st(a), as A varies among the definable subsets of n. As a corollary of a theorem in [Baisalov-Poizat, 1998], (Baisalov and Poizat) R is o-minimal. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Measure and Standard Part Standard Part and O-Minimal Lebesgue Measure Let L n be the pullback of L n via the map st. L n is an outer measure on Fin() n. A Fin() n is measurable iff 1 st(a) is measurable, and 2 st(a) st(fin() n \ A) is negligible. Let A n be bounded and definable. By [Baisalov-Poizat, 1998], st(a) is Borel, and a fortiori measurable. [Berarducci-Otero, 2004] prove that st(a) st(fin() n \ A) is negligible. Therefore, A is L n -measurable. L n (A) is the measure of A according to [Berarducci-Otero, 2004]. Measure and Standard Part Hausdorff Measure of the Standard Part When it does not work The above construction does not work for H d,n instead of L n. Let d < n, and H d st be the outer measure on Fin() n induced by H d,n via st. Example Let A the d-dimensional unit cube in d n. A is not H d st-measurable, because st(ã) is not Hd -negligible. For instance, if A is the translate of A by an infinitesimal vector A := { (x 1,..., x d, ε, 0,..., 0 : x i, 0 x i 1 }, then H d st(a A ) = 1 < H d st(a) + H d st(a ). A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

5 Measure and Standard Part The Grassmannian Hausdorff Measure of the Standard Part When it does work The Grassmannian Let C be a bounded subset of n, definable via a semi-algebraic map without parameters, and of dimension d. The restriction H d st to C is an outer measure. Any definable subset of C is H d st-measurable. If A C is definable via a semi-algebraic map without parameters, then H d st(a) = H d( A(R) ). Proof. Work in local charts. Grassmannian G(d, n) the set of d-dimensional R-linear subspaces of R n. G(d, n; ) the set of d-dimensional -linear subspaces of n. There is a compact manifold structure on G(d, n), of dimension (n d)d, such that G(d, n) SO(n)/ ( SO(d) SO(n d) ). Example G(1, n) G(n 1, n) P n 1 (R), the projective space. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 The Grassmannian The Grassmannian The Affine Grassmannian Haar Measure on G(d, n) and A(d, n) The affine Grassmannian A(d, n) is the set of (n d)-dimensional R-affine subspaces of R n, and similarly for A(d, n; ). A(d, n) is a fibred space over G(d, n), with fibre R d. To every affine (n d)-space E we associate the pair ( E, x ), where E G(d, n) is the d-linear space orthogonal to E; {x} = E E. A(d, n) can be also given a manifold structure, of dimension (n d)d + d. Example A(1, 2) is isomorphic to the Möbius band. The group SO(n) of linear isometries on R n acts naturally on G(d, n). There is a unique invariant measure µ G on G(d, n). The group of isometries of R n acts on A(d, n), and there is a corresponding invariant measure µ A on A(d, n). µ A is the product of µ G with L d. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

6 Integral-Geometric Measure Let C R n be a d-dimensional manifold. Then, there exists a constant 1 β = β(d, n) R >0 such that H d,n (C) = 1 #(C q) dµ A (q) = β A(d,n) = 1 (1) f C dµ G, β G(d,n) where f C (E) is the L E -area of p E (C), counted with multiplicity, with p E the orthogonal projection onto E. In general, the function I d,n (C) := 1 #(C q) dµ A (q) β A(d,n) is defined for any Borel set C, and I d,n is a Borel measure. 1 β = Γ ( ) ( d+1 Γ n d+1 ) ( Γ n+1 ) A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Example Integral-Geometric Measure d = n : A(n, n) = R n, and (1) becomes a tautology. d = 0 : A(0, n) is a singleton, H 0 (C) = #(C), and (1) is obvious. d = 1, n = 2 : For every θ [0, 2π), let D θ be the line forming an angle θ with the x-axis, N(r, θ) be the number of points on the curve C whose orthogonal projection on D θ is the point (r cos θ, r sin θ). Define P(θ) := + Then, the length of C is equal to I 1,2 (C) = 1 4 2π 0 N(r, θ) dr. P(θ) dθ. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Integral-Geometric Measure and O-Minimality Integral-Geometric Measure and O-Minimality Realisation of Grassmannian Definition There exist a semi-algebraic realisation of G(d, n) in R m (for some m) such that µ G coincides with the restriction of H k to G(d, n) (where k = dim ( G(d, n) ) ). Similarly for A(d, n). Therefore, the measure µ G induced by µg via the map st is an outer measure on G(d, n; ), such that definable subsets of G(d, n; ) are measurable. Similarly for A(d, n). Let C be a bounded definable subset of n. Define I d,n (C) := 1 #(C q) dµ A β (q) = A(d,n;) = 1 f C dµ G β, G(d,n;) where f C (E) is the L E -area of p E (C), counted with multiplicity. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

7 Basic Properties Integral-Geometric Measure and O-Minimality I d,n is a measure defined on all bounded definable subsets of n. Given C n definable and bounded, let S := { q A(d, n; ) : q C } S := { q A(d, n; ) : #(q C) = } S S 0 := S \ S. S, S and S 0 are bounded and definable. If dim(c) d, then dim(s ) < k := dim ( A(d, n; ) ). If dim(c) < d, then dim(s) < k. Therefore, if dim(c) d, then I d,n (C) <, and if dim(c) < d, then I d,n (C) = 0. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 More Properties Integral-Geometric Measure and O-Minimality Let C n be bounded, definable, with dim(c) d, and ( C(t) ) t be a uniformly bounded definable family of subsets of n, with dim ( C(t) ) d. 1 I d,n is invariant under rotations and translations. 2 For every r Fin() n, I d,n (rc) = st(r)d I d,n (C). 3 If C is defined by a semi-algebraic formula without parameters, then I d,n (C) = Id,n( C(R) ) = I d,n( st(c) ). ( ) C(t) is uniformly bounded. 4 I d,n 5 If is either an elementary extension or an o-minimal expansion of, then I d,n (C) = I d,n (C). 6 If d = n, I n,n (C) = Ln (C). 7 If C E, where E n is a d-dimensional plane, then I d,n (C) = LE (C) = H d,n( st(c) ), where L E is the o-minimal Lebesgue measure on E. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Integral-Geometric Measure and O-Minimality Outline Let n n, and C n be bounded and definable. Question I d,n (C) = Id,n (C)? 2 for the Area Lebesgue Measure, Integral, and Definable Functions Lipschitz and Bi-Lipschitz Functions Rectifiable Sets and Hausdorff Measure More Properties of Definable Functions: Regular Points, Sard s Lemma, Implicit Function Integral Geometry on O-Minimal Structures: Area, Coarea and e Open Problems: Area, General Area-Coarea, General Cauchy-Crofton Formulae A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

8 Area and Jacobian Lebesgue Measure, Integral, and Definable Functions Let f : R n R m be differentiable at a R n. The n-jacobian of f at a is J n (f )(a) := det ( Df (a) Df (a) T). 1. For every k it is possible to define J k (f )(a) as the square root of the sum of the square of the k minors of Df. 2. J k (f )(a) is the maximum k-dimensional volume of the image under Df (a) of a unit k-dimensional cube. 3. If k = m or n, ( J k (f ) ) 2 is the determinant of the k k product of Df with its transpose. The m-jacobian of A is J m (f )(a) := det ( Df (a) T Df (a) ). Note that J n (f )(a) = 0 iff rk ( Df (a) ) < n, and similarly for m. If n m and f is a smooth embedding of A R n into R m, then the area of f (A) is given by H n,m( f (A) ) = J n (f ) dl n. A A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Monads and Definable Functions Definition Lebesgue Measure, Integral, and Definable Functions Given x, y n, x and y are in the same monad, x y, iff x y is infinitesimal. 1. On Fin() n, x y iff st(x) = st(y). 2. Sf is definable in R. Let f : n m be definable. Definition Given c n, f preserves the monad of c iff For every x n, define x c f (x) f (c). Sf : R n R m { st ( f (x) ) iff f preserves the monad of x st(x) 0 otherwise. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

9 Lebesgue Measure, Integral, and Definable Functions Lebesgue Measure and Definable Functions Call I := [0, 1], and I R := [0, 1] R. Let f : I n Im be definable. Then, there exists C I n R definable and negligible such that, outside C, 1 f preserves the monads; 2 f is C 1 ; 3 Sf is C 1 ; 4 D(Sf ) = S(Df ). If moreover f preserves the monads on all I n, then Sf is continuous on all I n R. 1. Outside C can mean outside st 1 (C). 2. A proof of 1: let Γ be the graph of f, and Γ := st(γ). Since dim( Γ) dim(γ) = n, Γ is, outside a negligible set, the graph of a function. 3. The hypothesis can be weakened to f : I n m definable and almost bounded. 4. We use the saturation property for the continuity of Sf. 5. The latter is only a curiosity: we will not use it. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Lebesgue Measure, Integral, and Definable Functions Integral of Definable Functions Let f : I n I be definable, g := Sf : I n R I R, and [0, f ] := { (x, y) I n I : 0 y f (x)) } be the sub-graph of f. The following are well-defined and equal: I n I n R g dl n st(f ) dl n (L n L1 ) ( [0, st(f )] ) ( ) [0, f ] L n+1 L n+1( st ( [0, f ] )) L n+1( [0, g] ). 1. st(f ) : I n I R. 2. The hypothesis can be weakened to f : I n bounded. definable and almost Definition We define I n f dl n = f (x) dx := g dl n. I n I n R A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

10 Lipschitz and Bi-Lipschitz Functions Lipschitz and Bi-Lipschitz Functions Torricelli s Lipschitz and Bi-Lipschitz Functions Let A m and f : A n be definable. Definition f is Lipschitz iff there exists a rational number r > 0 such that x, y n f (x) f (y) r x y. The infimum of such constants r is Lip f, the Lipschitz constant of f. Note that if f is and C 1, and Df < r, then Lip f < r. Conversely, if f is Lipschitz and Lip f < r, then Df < r, where it is defined. Definition A definable bijection f : A B is bi-lipschitz iff both f and f 1 are Lipschitz. Let f : I I be definable and Lipschitz, and ḟ := df dx. Then, Example 1 0 ḟ (x) dx = st ( f (1) f (0) ). Let 0 < ε be infinitesimal. Define { x/ε if 0 x ε, Θ ε := 1 if ε x 1. Then, 1 0 Θ ε (x) dx = 0, while st ( Θ ε (1) Θ ε (0) ) = 1. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Lipschitz and Bi-Lipschitz Functions Lipschitz and Bi-Lipschitz Functions Fubini s Bi-Lipschitz Change of Variable For every n, m N, the (completion of the) measure L n+m is equal to the (completion of the) product L n Lm. Let f : I n+m I n+m I be definable. Then, f (x, y) dl m+n (x, y) = f (x, y) dl n (x) dlm (y). I m I n Let f : I n In be definable and bi-lipschitz, and A In be definable. Then, L n ( ) f (A) = det(df ) dl n. A If moreover g : I n I is definable, then g(x) dx = g ( f (y) ) det ( Df (y) ) dy. f (A) A A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

11 Rectifiable Sets Rectifiable Sets and Hausdorff Measure Let d N. Definition (Rectifiable sets) A set C n is basic d-rectifiable iff there exists A I d definable and open and f : A C definable bi-lipschitz bijection. (Decomposition into basic rectifiable sets) Let C I n definable and of dimension d. Then, there exist C 0,..., C m I m such that 1 C 0,..., C m are disjoint; 2 C 0 C m = C; 3 dim(c 0 ) < d; 4 each C i is basic d-rectifiable, i = 1,..., m. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Hausdorff Measure on Rectifiable Sets Definition Rectifiable Sets and Hausdorff Measure Let A I d be definable, f : Id In be a bi-lipschitz map, and C := f (A). Define H d,n (C) := J d (f ) dl d. A The quantity H d,n (C) does not depend on the particular choice of A and of f. Proof. By the bi-lipschitz change of variables. Lemma On a d-rectifiable set C Fin() n, H d,n is a eisler measure. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / In particular, C in the definition is basic d-rectifiable. 2. Say what a eisler measure is. Rectifiable Sets and Hausdorff Measure Change of Variables for Hausdorff Measure Lemma Let B I n and C Im be basic d-rectifiable, and f : B C be a bi-lipschitz bijection. Then, J d (f ) dh d,n = Hd,m (C). B A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

12 Rectifiable Sets and Hausdorff Measure Rectifiable Sets and Hausdorff Measure of the Standard Part 1. Note that f is bi-lipschitz, and therefore f (A) is basic d-rectifiable. 2. It is possible to decompose any d-dimensional definable set C into sets C i satisfying the condition (namely the existence of a function f i as in the lemma, after a permutation of variables). Lemma (*) Let d < n and π : n d be the projection onto the first d coordinates. Let A I d be definable, and f : A In be definable and Lipschitz, such that π f = id A. Then, H d,n ( ) f (A) = H d,n ( st(f (A)) ) = H d,n( Sf (st(a)) ). A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 More Properties of Definable Functions More Properties of Definable Functions Regular Points Morse-Sard s Lemma Definition Let f : I n Im be a definable function, and g := Sf : In R Im R. An element c I n is a S-regular point for f iff there exists ε Q>0 such that, such that, if we call B the ball of centre c and radius ε, 1 B (0, 1) n R ; 2 f preserves the monads on B; 3 f is C 1 on B; 4 Df is bounded and preserves the monads on B; 5 g is C 1 on B; 6 S(Df ) = Dg on B; 7 every b B is a regular point for g. Let f : I n Im be definable, with m n. Let C I n be the sets of points c In such that 1 satisfy the conditions 1 6; 2 rk ( Df (c) ) = n. Then, H n,m ( f (I \ C) ) = 0. In particular, the set of S-singular values for f is L n -negligible. Note that every x st 1 (B) is a regular point for f, and that the set of S-regular points is open. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

13 1. We have no control over the case m < n, because the set of non-c 1 points in the domain I n can have dimension up to n The corollary is trivial if m > n, because the whole image of f has dimension at most n. Implicit Function More Properties of Definable Functions Let f : I n In be definable, g := Sf, a In R be a S-regular point for f, and c := g(a). Then, for every y st 1 (c) there exists a unique x st 1 (a) such that f (x) = y. Moreover, fix x 0 st 1 (a). Then, there exist ρ, r Q >0 such that, if then B := cl ( B ρ (a) ) and B := cl ( B ρ (x 0 ) ) st 1 (B), 1 f B and g B are open and injective, 2 B r (f (x 0 )) f (B ), 3 for every a B and y st 1( g(a ) ) there exists a unique x B such that f (x) = y. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Implicit Function Corollary More Properties of Definable Functions In the above theorem, let c I n be a S-regular value, and st(y) = c. Then, # ( g 1 (c) ) = # ( f 1 (y) ). Sketch of proof of the theorem. Consider the one of the usual proof of the implicit function theorem from standard calculus, via a contraction T : B B. Adapt it to the o-minimal situation. Verify that T is definable, and that the various constants can be chosen non-infinitesimal for S-regular points. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Aside: the Image of a Measurable Set More Properties of Definable Functions Let f : I n In be definable and preserving the monads. Let X In be -measurable. Then, L n 1 f (X ) is also L n -measurable; 2 if X is L n -negligible, then f (X ) is also Ln -negligible. Sketch of proof. Let A := st(x ), Y := f (X ), and g := Sf. Then, 1 A is L n -measurable; 2 g(a) = st(y ); 3 st(y ) is L n -measurable; 4 if X is negligible, then g(a) is also negligible, and therefore Y is negligible. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

14 More Properties of Definable Functions Integral Geometry and O-Minimality Counter-Example Coarea Formula Example Let 0 < ε be infinitesimal. Define { x/ε if 0 x ε, Θ ε := 1 if ε x 1. Let Y [0, 1 2 ] be non Ln -measurable, and X := Θ 1 ε (Y ) [0, ε]. Then, L n (X ) = 0, but Θ ε (X ) = Y is non L n -measurable. Let f : I m In be a definable Lipschitz function, with m > n, and A I m be a definable set. Then, J n f dl m = H m n,m ( A f 1 (y) ) dl n (y). A I n A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Integral Geometry and O-Minimality Integral Geometry and O-Minimality Coarea Formula Area Formula Particular case Sketch of proof. 1 By Morse-Sard s lemma, we can assume that all points are S-regular. 2 Apply the Coarea formula to g := Sf and B := st(a), obtaining A J n f dl m = B J n g dl m = I n R H m n,m( B g 1 (z) ) dl n (z). 3 By the implicit function theorem, and Lemma (*), for almost every y I n, we have H m n,m ( A f 1 (y) ) = H m n,m( B g 1 (st(y)) ). Let f : I n In be a definable and Lipschitz function. Then, det Df dl n = #(f 1 (y)) dl n (y) I n If moreover h : D n D is definable, then h(x) det Df (x) dl n (x) = I n I n I n x f 1 (y) h(x) dl n (y). A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

15 Integral Geometry and O-Minimality Integral Geometry and O-Minimality Area Formula Lemma Sketch of proof. By Morse-Sard s lemma, we can assume that all points are S-regular. Apply the corollary to the implicit function theorem: Let A I n be a basic d-rectifiable set, E G(d, n; ), and LE be the Lebesgue measure on E Then, for L E -almost every q E, we have #(q A) = # ( st(q) st(a) ). c y st 1 (c) Apply the area formula to g. #(f 1 (y)) = #(g 1 (c)). (Cauchy-Crofton) Let f : I m In be a definable Lipschitz function, with m n, and C I m be a definable set. Then, H d,n (C) = Id,n (C) = 1 #(q C) dµ A β = 1 f C dµ G A(d,n;) β. G(d,n;) A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / L E coincides with the H d,n -measure restricted to E. 2. To prove the lemma, apply the implicit function theorem and Morse-Sard s lemma to the orthogonal projection onto E, compose with a bi-lipschitz bijection f : I d A. Sketch of proof. Integral Geometry and O-Minimality Call B := st(c). We can assume that C is basic d-rectifiable and that H d,n (B) = H d,n (C). By the Lemma, #(q C) dl E (q) dµ G (E) = E G(d,n;) q E E G(d,n;) F G(d,n) p st(e) p F #(p B) dl st(e) (p) dµ G (E) = #(p B) dl F (p) dµ G (F ) = H d,n st(b), where we used the Cauchy-Crofton formula for the last equality. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

16 More Properties Integral Geometry and O-Minimality Let C n be bounded, definable, with dim(c) d, and ( C(t) ) t be a uniformly bounded definable family of subsets of n, with dim ( C(t) ) d. Integral Geometry and O-Minimality 1 I d,n is invariant under rotations and translations. 2 For every r Fin() n, I d,n (rc) = st(r)d I d,n (C). 3 If C is defined by a semi-algebraic formula without parameters, then I d,n (C) = Id,n( C(R) ) = I d,n( st(c) ). ( ) C(t) is uniformly bounded. 4 I d,n 5 If is either an elementary extension or an o-minimal expansion of, then I d,n (C) = I d,n (C). 6 If d = n, I n,n (C) = Ln (C). 7 If C E, where E n is a d-dimensional plane, then I d,n (C) = LE (C) = H d,n( st(c) ), where L E is the o-minimal Lebesgue measure on E. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 8 Let n n, and C n be bounded and definable. Then, H d,n (C) = Hd,n (C). A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Open Problems Open Problems Open Problems Conjecture (Area formula) Let f : I m In be a definable Lipschitz function, with m n. 1 If A I m is a Lm -measurable set, then J m (f ) dl m = #(f 1 (y) A) dh m (y). A 2 If g : I m I is a definable function, then g J m (f ) dl m = I n I m I n x f 1 (y) g(x) dh m (y). Conjecture (General Area-Coarea formula) Let W be a subset definable of I n of dimension m, and Z a definable subset of I ν of dimension µ, with m µ 1. Let f : W Z be a definable Lipschitz function. 1 If A I m is definable, then J µ (f ) dh m = A Z H m µ ( f 1 (z) ) dh µ (z). 2 If g : W I is a definable function, then g J µ (f ) dh m = g dh m µ dh µ (z). W Z f 1 (z) A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57

17 Open Problems Summary Summary Conjecture (General Cauchy-Crofton formula) Let 0 k m n, and d := m k. Let A I n be definable and of dimension m. Then, H m (A) = 1 H k (A q) dhn d β (q), q A(n d,n;) where β = β(k, m, n) is a constant not depending on or A. We can define the Hausdorff measure on o-minimal structures. This measure behaves as expeted, with respect to definable Lipschitz maps. In particular, it satisfies the formulae of Cauchy-Crofton, change of variables and coarea. Some other properties are only conjectured. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57 Bibliography Bibliography P. R. Halmos. Measure Theory. D. Van Nostrand Company, Inc., New York, N. Y., F. Morgan. Geometric Measure Theory. Academic Press, 1988 An introduction to Federer s book by the same title. Y. Baisalov and B. Poizat. Paires de structures o-minimales. J. Symbolic Logic, 63(2): , O-minimality of the standard part. A. Berarducci and M. Otero. An additive measure in o-minimal expansions of fields. Q. J. Math., 55(4): , Lebesgue measure on o-minimal structures. A. Fornasiero (Università di Pisa) Hausdorff Measure Paris / 57