Hausdorff measure on o-minimal structures

Version 3.6

A. Fornasiero & E. Vasquez Rifo

November 17, 2010

Abstract

We introduce the Hausdorff measure for definable sets in an ominimal structure, and prove the Cauchy-Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

Keywords: O-minimality, Hausdorff measure, Whitney arc property, Cauchy-Crofton, coarea.

MSC 2010 : Primary: 03C64; Secondary: 28A75.

1 Introduction

Let K be an o-minimal structure expanding a field. We introduce, for every e ∈ ℕ, the e-dimensional Hausdorff measure for definable sets, which is the generalization of the usual Hausdorff measure for real sets [Morgan88]. We also show that every definable set can be partitioned into “basic e-rectifiable sets” (§3). Moreover, we generalize some well known result from geometric measure theory, such as the Cauchy-Crofton formula (which computes the Hausdorff measure of a set as the average number of points of intersection with hyperplanes of complementary dimension) and the co-area formula (a generalization of Fubini’s theorem), to the o-minimal context.

The measure defined in [BO04] is the starting point for our construction of the Hausdorff measure. A theorem of [BP98] allows us to prove that integration using the Berarducci-Otero measure satisfies properties analogous to the ones for integration over the reals (for example, the change of variable formula). If K is sufficiently saturated, the Berarducci-Otero measure of a bounded definable set X is L_{ℝ}(st(X)), where L_{ℝ} is the Lebesgue measure and st is the standard-part map. However, the naive definition of Hausdorff measure given by

H^{e}(X) := H_{ℝ}^{e}(st(X))  (1)

does not work (because the resulting “measure” is not additive: see Example 5.8). The correct definition for the e-dimensional Hausdorff measure is defining it first for basic e-rectifiable sets via (1), and then extending it to definable sets by using a partition into basic e-rectifiable pieces. Such a partition is obtained by using partitions into M_{n}-cells ([K92], [P08], [VR06]), a consequence of which is the Whitney arc property for basic e-rectifiable sets (§4).

2 Lebesgue measure on o-minimal structures

The definitions of measure theory are taken from [Halmos50].

Let ℝ̄ := ℝ ∪ {±∞} be the extended real line. Let K be a ℵ_{1}-saturated o-minimal structure, expanding a field. Let K̊ be the set of finite elements of K. Let st : K^{n} → ℝ̄^{n} be the function mapping x̄ to the n-tuple of standard parts of the components of x̄.

For every n ∈ ℕ, let L_{ℝ}^{n} be the n-dimensional Lebesgue measure (on ℝ^{n}). If n is clear from context we drop the superscript. Let L_{1}^{n} be the o-minimal measure on K̊^{n} defined in [BO04]. More precisely, L_{1}^{n} is a measure on the σ-ring R_{n} generated by the definable subsets of K̊^{n}; thus, (K̊^{n}, R_{n}, L_{1}^{n}) is a measure space. Moreover, since K̊^{n} ∈ R_{n}, R_{n} is actually a σ-algebra.

Notice that L_{1}^{n} can be extended in a natural way to a measure L_{2}^{n} on the σ-ring B_{n} generated by the definable subsets of K^{n} of finite diameter. Finally, we denote by L^{n} the completion of L_{2}^{n}, and if n is clear from context we drop the superscript. Notice that the σ-ring B_{n} is not a σ-algebra.

Remark 2.1 ([BO04, Thm. 4.3]). If C ⊂ K̊^{n} is definable, then L^{n}(C) is the Lebesgue measure of st(C) .

Definition 2.2. For A ⊆ K^{n} and f : K^{n} → K^{m} we define st(f) : A → ℝ̄^{m} by st(f)(x) = st(f(x)).

Remark 2.3. If A ⊆ K̊^{n} and f : A → K are definable, then st(f) is an L^{n}-measurable function.

Definition 2.4. Let A ⊆ K̊^{n} and f : A → K be definable. If st(f) is L^{n}-integrable we will denote its integral by

∫_{A} f dL^{n}; ∫_{A} f(x) dx; ∫_{A} f(x) dL^{n}(x) or ∫_{A} f.

Remark 2.5. If A ⊆ K̊^{n} and f : A → K̊ are definable, then st(f) is L-integrable.

Let ℝ_{K} be the structure on ℝ generated by the sets of the form st(U), where U varies among the definable subsets of K^{n}. By [BP98], ℝ_{K} is ominimal.

Remark 2.6. Let U ⊆ K̊^{n} be definable. Then, dim(st(U)) ≤ dim(U).

Proof. Let dim(U) = d. After a cell decomposition, we can assume that U is the graph of a definable continuous function f : V → K̊^{n−d}, with V ⊂ K̊^{d} open cell. We can then conclude by applying the method in [HPP08, Lemma 10.3].

Definition 2.7. A function f is Lipschitz if there is C ∈ K̊ such that, for all x, y ∈ dom(f), we have |f(x) − f(y)| < C|x − y| (notice the condition on C being finite). An invertible function f is bi-Lipschitz if both f and f^{−1} are Lipschitz.

Remark 2.8. Let U ⊂ K̊^{n} and f : U → K̊ be definable, with f ≥ 0. Then,

∫_{U} f dL^{n} = L^{n+1}({⟨x̄, y⟩ ∈ U × K : 0 ≤ y ≤ f(x̄)}).

Lemma 2.9 (Change of variables). Let U, V ⊆ K̊^{n} be open and definable, and let A ⊆ U be definable. Let f : U → V be definable and bi-Lipschitz and g : V → K̊ be definable, then

∫_{f(A)} g = ∫_{A} |det Df| g ∘ f.

Before proving the above lemma, we need some preliminary definitions and results.

Lemma 2.10. Let U ⊂ K̊^{n} be open and let f : U → K̊ be definable. Then there is a ℝ_{K}-definable function \overline{f} : C → ℝ, where C ⊂ st(U) is an open set, such that

i) E := (st(U) \ C) ∪ (C ∩ st(K^{n} \ U)) is L_{ℝ}^{n}-negligible (and, therefore, st^{−1}(E) is L^{n}-negligible).

ii) f and \overline{f} are C^{1} on U \ st^{−1}(E) and C, respectively.

iii) For every x ∈ U with st(x) ∈ C we have st(f(x)) = \overline{f}(st(x)). Moreover, Df is finite and D(\overline{f})(st x) = st(Df(x)).

∫_{U} f = ∫_{C} \overline{f}.

Proof. By cell decomposition, we may assume that f is a function of class C^{1}, and that U is an open cell. Since dim(Γ(f)) = n, we have, by Remark 2.6, dim(st(Γ(f)) ≤ n. By cell decomposition, there is an ℝ_{K}-definable, closed, negligible set E ⊂ st(U), and definable functions g_{k} : st(U) \ E → ℝ of class C^{1} for k = 1, …, r such that st(Γ(f)) ∩ ((st(U) \ E) × ℝ) is the union of the graphs of the functions g_{i}. We claim that r = 1: In fact, if g_{1}, g_{2} are two different such functions, and say g_{1} < g_{2}, then for some x ∈ st(U) we have ⟨x, g_{1}(x)⟩, ⟨x, g_{2}(x)⟩ ∈ st(Γ(f)). Since f is continuous,{⟨x, y⟩ : y ∈ (g_{1}(x), g_{2}(x))} ⊂ st(Γ(f)). On the other hand, {⟨x, y⟩ : ⟨x, y⟩ ∈ st(Γ(f))} is the finite set {⟨x, g_{1}(x)⟩, …, ⟨x, g_{r}(x)⟩}, absurd.

By [HPP08, Theorem 10.4], after enlarging E by a negligible set, we obtain i).

Let \overline{f} := g_{1}. ii) holds, and for every x ∈ U with st(x) ∈ C we have st(f(x)) = \overline{f}(st(x)). The equality of the integrals in iv) follows from Remark 2.8. To obtain the second part of iii) we will enlarge E by a negligible set. For i = 1, …, n let

E_{i} := st(\{x ∈ U : \frac{∂f}{∂x_{i}} (x) ∉ K̊\}).

By [BP98], E_{i} is ℝ_{K}-definable. If dim(E_{i}) = n, then E_{i} contains an open ball. This contradicts Lemma 2.5 of [BO04] by which every definable, one variable function into K̊ has finite derivative except on st^{−1}(A), for a finite set A. It follows that each set E_{i} is negligible and therefore, after enlarging E, we may assume that D(f) is finite on U \ st^{−1}(E).

It remains to prove D(\overline{f})(st x) = st(Df(x)). As before, we will enlarge E by a negligible set. Let V := \{x ∈ ℝ^{n} : D(\overline{f})(x) ≠ \overline{Df}(x)\}. The set V is ℝ_{K}-definable. If V is non-negligible, then it contains an open ball and therefore w.l.o.g. we may assume that V is an open ball centered at 0. We may also assume f(0) = 0. After substracting from f a linear function, we can assume that \frac{∂f}{∂x_{i}} (0) = 0 and \frac{∂\overline{f}}{∂x_{i}} (0) = 3ϵ > 0 for some index i = 1, …, n. Therefore, on a smaller neighborhood of 0, we have \frac{∂f}{∂x_{i}} < ϵ and

\frac{∂\overline{f}}{∂x_{i}} > 2ϵ. Thus, for x along the x_{i} axis, |f(x)| < |x|ϵ and \overline{f}(x) ≥ 2|x|ϵ contradicting the first part of iii), namely, st(f(x)) = \overline{f}(x). We conclude that V is negligible. Let E′ be a negligible set such that away from st^{−1}(E′) the equality st(Df(x)) = \overline{Df}(st x) holds. Then away from st^{−1}(V ∪ E′) we have st(Df(x)) = \overline{Df}(st(x)) = D\overline{f}(st(x)) as wanted. By cell decomposition, E can be further enlarged so that C is open.

Remark 2.11. If f^{−1}(A) is negligible whenever A is, then, outside a negligible closed set, \overline{(f ∘ g)} = \overline{f} ∘ \overline{g}.

Proof of Lemma 2.9. The fact that f is bi-Lipschitz implies that \overline{f} is injective (since it is also bi-Lipschitz). Claim 1. Let C ⊂ st(V) be Lebesgue measurable. Then,

L^{n}(C) = ∫_{(st f)^{−1}(C)} st(|det Df|).

In fact, by the change of variables formula (on the reals!) and Lemma 2.10,

L^{n}(C) = ∫_{\overline{f}^{−1}(C)} |det D\overline{f}|) = ∫_{(st f)^{−1}(C)} st(|det Df|).

Claim 2. Let h : V → ℝ̄ be an integrable function. Then,

∫_{V} h = ∫_{U} st(|det Df|) h ∘ f.

Claim 1 implies that the statement is true if h is a simple function. By continuity, the statement is true for any integrable function h.

In particular, we can apply Claim 2 to the function

h : x ↦ \begin{cases}st(g(x)) & if x ∈ f(A), \\ 0 & otherwise, \end{cases}

and obtain the conclusion.

Lemma 2.12 (Fubini’s theorem). L^{n+m} is the completion of the product measure L^{n} × L^{m}. Therefore, if D is the interval [0, 1] ⊂ K and given f : D^{n+m} → D definable,

∫_{D^{n+m}} f(x, y) dL^{n+m}(x, y) = ∫_{D^{m}}∫_{D^{n}} f(x, y) dL^{m}(x) dL^{n}(y).

Proof. Follows from the definition of L^{n} in [BO04].

2.1 Measure on semialgebraic sets

Definition 2.13. We say that E ⊆ K^{n} is ∅-semialgebraic if E is definable without parameters in the language of pure fields. If E ⊆ K^{n} is ∅- semialgebraic we denote the subset of ℝ^{n} defined by the same formula that defines E by E_{ℝ}.

Remark 2.14. Let E ⊆ K̊^{n} be ∅-semialgebraic. Then, st(E) = \overline{E_{ℝ}}.

Let E ⊆ K^{n} be closed and ∅-semialgebraic submanifold. Working in local charts, from [BO04] one can easily define a measure L^{E} on the σ-ring generated by the definable subsets of E of bounded diameter. We will denote in the same way the completion of L^{E}. Notice that L^{K^{n}} = L^{n}.

Remark 2.15. Let E be a closed, ∅-semialgebraic submanifold of K^{n} of dimension e, F := st(E), and C ⊆ E be definable and bounded. Then, L^{E}(C) = L_{ℝ}^{F}(st(C)), where L_{ℝ}^{F} is the e-dimensional Hausdorff measure on F.

One could also take the above remark as the definition of L^{E} on E ∩ K̊^{n}.

3 Rectifiable partitions

Theorem 3.8 shows that every definable set A ⊂ K̊^{n} has a partition into definable sets which are M_{n}-cells after an orthonormal change of coordinates (where M_{n} ∈ ℚ depends only on n). In [P08], the author shows that a permutation of the coordinates suffices. The proof of 3.8 follows closely that of [K92]. The partition in 3.8 is then used in Corollary 3.11 to show that definable sets have a rectifiable partition.

Definition 3.1. Let L : V → W be a linear map between normed K-vector spaces. The norm of L is given by

∥L∥ := sup_{|v|=1}|L(v)|.

For V, W in the Grassmannian of e-dimensional linear subspaces of K^{n}, namely G_{e}(K^{n}), let π_{V} and π_{W} ∈ End_{K}(K^{n}) be the orthogonal projections onto V and W respectively. In this way we have a canonical embedding G_{e}(K^{n}) ⊂ End_{K}(K^{n}). The distance function on the Grassmannian is given by the inclusion above:

δ(V, W) := ∥π_{V} − π_{W} ∥.

For P in G_{1}(K^{n}) and X ∈ G_{k}(K^{n}), define

δ(P, X) := |v − π_{X}(v)|,

where π_{X} is the orthogonal projection onto X, and v is a generator of P of norm 1. Note that δ(P, X) = 0 if and only if P ⊂ X, 0 ≤ δ(P, X) ≤ 1 and δ(P, X) = 1 if and only if P ⊥ X. Note also that δ(P, X) is the definable analogous of the sine of the angle between P and X.

Lemma 3.2. Let n ∈ ℕ_{>0}. Then there exists an ϵ_{n} ∈ ℚ_{>0}, ϵ_{n} < 1, such that for any X_{1}, …, X_{2n} ∈ G_{n−1}(K^{n}), there is a line P ∈ G_{1}(K^{n}) such that whenever Y_{1}, …, Y_{2n} ∈ G_{n−1}(K^{n}) and

\begin{aligned}δ(X_{i}, Y_{i}) & < ϵ_{n}, i = 1, …, 2n, then \\ δ(P, Y_{i}) & > ϵ_{n}, i = 1, …, 2n. \end{aligned}

Proof. For ϵ > 0 define S_{i}(ϵ) = {v ∈ S^{n−1} : |v − π_{X_{i}}(v)| ≤ 2ϵ}. If K = ℝ, let ϵ_{n} ∈ ℚ_{>0} be small enough so that 2n Vol(S_{1}(ϵ_{n})) < Vol(S^{n−1}), where Vol is the measure L^{S^{n−1}} defined in §2.1. Then

Vol(⋃_{i=1}^{2n} S_{i}(ϵ_{n})) ≤ 2n Vol(S_{1}(ϵ_{n})) < Vol(S^{n−1})

and therefore

⋃_{i=1}^{2n} S_{i}(ϵ_{n}) ≠ S^{n−1}.  (2)

The same ϵ_{n} will necessarily satisfy (2) for any field K containing ℝ.

Now, we choose

v ∈ S^{n−1} − ⋃_{i=1}^{2n} S_{i}(ϵ_{n})

and let P := ⟨v⟩. Then

δ(P, Y_{i}) = |v − π_{Y_{i}}v| ≥ |v − π_{X_{i}}v| − |π_{X_{i}}v − π_{Y_{i}}v| > ϵ_{n}.

Definition 3.3. Let ϵ > 0. A definable embedded submanifold M of K^{n} is ϵ-flat if for each x, y ∈ M we have δ(T M_{x}, T M_{y}) < ϵ, where T M_{x} denotes the tangent space to M at x.

Lemma 3.4. Let A ⊂ K^{n} be a definable submanifold of dimension e and ϵ ∈ ℝ_{>0}. Then there is a cell decomposition A = ⋃_{i=0}^{k} A_{i} of A such that for every i we have either dim(A_{i}) < dim(A) or A_{i} is an ϵ-flat submanifold of K^{n}.

Proof. Cover G_{e}(K^{n}) by a finite number of balls B_{i} of radius ϵ/2; and consider the Gauss map G : A → G_{e}(K^{n}) taking an element a of A to T A_{a}. Take a cell decomposition of K^{e} compatible with A and partitioning each G^{−1}(B_{i}). Then the e-dimensional cells contained in A are ϵ-flat.

Lemma 3.5. Let ϵ ∈ ℚ_{>0}, and let A ⊂ K̊^{n} be an open definable set. Then there are open, pairwise disjoint cells A_{1}, …, A_{p} ⊂ A such that

(i) dim(A − ⋃ A_{i}) < n.

(ii) For each i, there are definable, pairwise disjoint sets B_{1}, …, B_{k} (with k depending on i) such that

(a) k ≤ 2n;

(b) each B_{j} is a definable subset of ∂A_{i} and an ϵ-flat, (n−1)-dimensional, C^{1}-submanifold of K^{n};

(c) dim(∂A_{i} − ⋃_{j=1}^{k} B_{j}) < n − 1.

Proof. By induction on n. The lemma is clear for n = 1. Assume that n > 1 and the lemma holds for smaller values of n.

Take a cell decomposition of \overline{A} compatible with A into C^{1}-cells. Let C be an open cell in this decomposition; it suffices to prove the lemma for C. Note that C = (f, g)_{X}, where X is an open cell in K^{n−1} and f, g are definable C^{1}-functions on X. Take finite covers of Γ(f) and Γ(g) by open, definable sets U_{i} and V_{j}, respectively, such that each U_{i} ∩ Γ(f) and each V_{j} ∩ Γ(g) is ϵ-flat (to do this, take a finite cover of the Grassmannian by ϵ-balls and pull it back via the Gauss maps for Γ(f) and Γ(g)). The collection of all sets π(U_{i}) ∩ π(V_{j}) is an open cover O of X. By the cell decomposition theorem, there is a C^{1}-cell decomposition of X partitioning each set in O. Let S be an open cell in this decomposition, and let C_{0} := (f, g)_{S}. It suffices to prove the lemma for C_{0}. By the inductive hypothesis, we can find A′_{1}, …, A′_{p} ⊂ S and B′_{1}, …, B′_{k} ⊂ ∂A′_{i} satisfying the conditions (i) and (ii) above (with n replaced by n − 1). Define

A_{i} := (f, g)_{A′_{i}}, i = 1, …, p.

Then dim(C_{0} − ⋃_{i=1}^{p} A_{i}) < n. For j = 1, …, k, the set (B′_{j} × K) ∩ ∂A_{i} is definable. Take a C^{1}-cell decomposition of this set, and let B_{j} be the union of the (n − 1)-dimensional cells in this decomposition (note that B_{j} may be empty). Then B_{j} is an ϵ-flat C^{1}-submanifold of K^{n} and

dim(((B′_{j} × K) ∩ ∂A_{i}) − B_{j}) < n − 1.

Define B_{k+1} := Γ(f|A′_{i}) and B_{k+2} := Γ(g|A′_{i}); by construction these are ϵ-flat. It is routine to see that ∂A_{i} ⊂ B_{k+1} ∪ B_{k+2} ∪ (∂A′_{i} × K). Thus

∂A_{i} − ⋃_{j=1}^{k+2} B_{j} ⊂ ((∂A′_{i} × K) ∩ ∂A_{i}) − ⋃_{j=1}^{k} B_{j}
= (⋃_{j=1}^{k}((B′_{j} × K) ∩ ∂A_{i}) ∪ E) − ⋃_{j=1}^{k} B_{j}
⊂ ⋃_{j=1}^{k}(((B′_{j} × K) ∩ ∂A_{i}) − B_{j}) ∪ E,

where E is a definable set with dim(E) < n − 1. Therefore dim(∂A_{i} − ⋃_{j=1}^{k+2} B_{j}) < n − 1. Since k ≤ 2(n − 1), we get k + 2 ≤ 2n and the lemma is proved.

Definition 3.6. Let U ⊆ K^{n} be open and let f : U → K^{m} be definable. Given 0 < M ∈ K, we say that f is an M-function if |Df| ≤ M. We say that f has finite derivative if |Df| is finite.

Notice that, by ω-saturation of K, if f is definable and has finite derivative, then it is an M-function for some finite M.

Let M ∈ K_{>0}. An M-cell is a C^{1}-cell where the C^{1} functions that define the cell are M-functions. More precisely:

Definition 3.7. Let (i_{1}, …, i_{m}) be a sequence of zeros and ones, and M ∈ K_{>0}. An (i_{1}, …, i_{m})-M-cell is a subset of K^{m} defined inductively as follows:

(i) A (0)-M-cell is a point {r} ⊂ K, a (1)-M-cell is an interval (a, b) ⊂ K, where a, b ∈ K.

(ii) An (i_{1}, …, i_{m}, 0)-M-cell is the graph Γ(f) of a definable M-function f : X → K of class C^{1}, where X is an (i_{1}, …, i_{m})-M-cell; an (i_{1}, …, i_{m}, 1)- M-cell is a set

(f, g)_{X} := {(x, r) ∈ X × K : f(x) < r < g(x)},

where X is an (i_{1}, …, i_{m})-M-cell and f, g : X → K are definable M-functions of class C^{1} on X such that for all x ∈ X, f(x) < g(x).

Theorem 3.8. Let A ⊂ K̊^{n} be definable. Then there are definable, pairwise disjoint sets A_{i}, i = 1, …, s, such that A = ⋃_{i} A_{i} and for each A_{i}, there is a change of coordinates σ_{i} ∈ O_{n}(K) such that σ_{i}(A_{i}) is an M_{n}-cell, where M_{n} ∈ ℚ_{>0} is a constant depending only on n.

Proof. We will make use of the following fact:

Let ϵ ∈ [0, 1], P ∈ G_{1}(K^{n}), X ∈ G_{k}(K^{n}) and and w ∈ X be a unit vector. Suppose δ(P, X) > ϵ. If π_{P} (w) ≥ 1/2, where π_{P} is the orthogonal projection onto P, then

|π_{P} (w) − w| ≥ |π_{P} (w) − π_{X}(π_{P} (w))| > |π_{P} (w)|ϵ ≥ 1/2ϵ.

If π_{P} (w) < 1/2, then |w| ≤ |π_{P} (w)| + |π_{p}(w) − w| ≤ 1/2 + |π_{p}(w) − w|. In either case, we have

|π_{P} (w) − w| ≥ \frac{1}{2}ϵ.  (3)

We prove the theorem by induction on n; for n = 1 the theorem is clear. We assume that n > 1 and that the theorem holds for smaller values of n. We also proceed by induction on d := dim(A). It’s clear for d = 0; so we assume that d > 0 and the theorem holds for definable bounded subsets B of K^{n} with dim(B) < d.

Case I: dim(A) = n. In this case A is an open, bounded, definable subset of K^{n}, so by using the inductive hypothesis and Lemma 3.5, we can reduce to the case where there are pairwise disjoint, definable B_{1}, …, B_{k} ⊂ ∂A such that k ≤ 2n, dim(∂A−⋃_{j=1}^{k} B_{j}) < n−1 and each B_{j} is an ϵ_{n}-flat submanifold, where ϵ_{n} is as in Lemma 3.2. By Lemma 3.2, there is a hyperplane L such that for each B_{j} and all x ∈ B_{j}, we have δ(L^{⊥}, T_{x}B_{j}) > ϵ_{n}. Take a cell decomposition B of K^{n}, with respect to orthonormal coordinates in the L, L^{⊥} axis, partitioning each B_{j}. Let

S := {C ∈ B : dim(C) = n − 1, C ⊂ ⋃_{j=1}^{k} B_{j}}

and note that dim(∂A \ ⋃_{C∈S} C) < n − 1. Furthermore,

BAD := {x ∈ A : π_{L}^{−1}(π_{L}(x)) ∩ ∂A ⊄ ⋃_{c∈S} C}

has dimension smaller than n. Let U_{1}, …, U_{l} be the elements of {π_{L}(C) : C ∈ S}. Then the set

{x ∈ A : x ∉ π_{L}^{−1}(⋃_{i=1}^{l} U_{i})}

is contained in BAD, and therefore has dimension smaller than n.

By using the inductive hypothesis, we only need to find the required partition for each of the sets A ∩ π_{L}^{−1}(U_{i}), i = 1, …, l. Fix i ∈ {1, …, l} and let U := U_{i}, A′ := A∩π_{L}^{−1}(U). Take C ∈ S with π_{L}(C) = U. Then C = Γ(ϕ) for a definable C^{1}-map ϕ : U → L^{⊥} and for all x ∈ C,

T_{x}C = {(v, Dϕ(v)) : v ∈ T_{π_{L}(x)}U}.

Let v ∈ T_{π_{L}(x)}U be a unit vector; since δ(L^{⊥}, T_{x}C) > ϵ_{n} and |(v, Dϕ(v))| = \sqrt{1 + |Dϕ(v)|^{2}}, it follows from equation (3) that

\frac{1}{2}ϵ_{n} ≤ \frac{1}{\sqrt{1 + |Dϕ(v)|^{2}}} |π_{L^{⊥}}((v, Dϕ(v))) − (v, Dϕ(v))| = \frac{1}{\sqrt{1 + |Dϕ(v)|^{2}}} |v|.

Therefore,

|Dϕ(v)| ≤ \sqrt{\frac{4}{ϵ_{n}^{2}} − 1}.

Let M_{n} ∈ ℚ be bigger than max \{M_{n−1}, \sqrt{\frac{4}{ϵ_{n}^{2}} − 1}\} .

We have proved that for each C_{j} ∈ S with π_{L}(C_{j}) = U there is a definable C^{1}-map ϕ_{j} : U → K, such that |Dϕ_{j}| < M_{n} and C_{j} = Γ(ϕ_{j}).

By the inductive hypothesis, there is a partition P of U such that each piece P ∈ P is a M_{n−1}-cell after a change of coordinates of L. We have

A′ = ∐_{\begin{matrix}P ∈P \\ (ϕ_{r},ϕ_{s})_{P} ⊂A′ \end{matrix}}(ϕ_{r}, ϕ_{s})_{P} ,

and (ϕ_{r}, ϕ_{s})_{P} is a M_{n}-cell after a coordinate change.

Case II: dim(A) < n. In this case, by Lemma 3.4, we can partition A into cells which are ϵ_{n}-flat. Therefore we may assume that A is an ϵ_{n}-flat submanifold, where ϵ_{n} is as in Lemma 3.2. As in case I, there is a hyperplane L such that A is the graph of a function f : U → K, U ⊂ L and |Df| < M_{n}. By the inductive hypothesis, we can partition U into M_{n−1}-cells. The graphs of f over the cells in this partition give the required partition of A.

Definition 3.9. Let A ⊆ K^{n} and e ≤ n. A is basic e-rectifiable with bound M if, after a permutation of coordinates, A is the graph of an Mfunction f : U → K^{n−e}, where U ⊂ K^{e} is an open M-cell for some finite M.

Lemma 3.10. Let A ⊂ K̊^{n} be an M-cell of dimension e. Then, A is a basic e-rectifiable set, and the bound of A can be chosen depending only on M and n.

Proof. We proceed by induction on n. If n = 0 or n = 1 the result is trivial, so assume n ≥ 2. By definition, there exists an M-cell B ⊂ K̊^{n−1} such that

(1) either A = Γ(g) for some M-function g : B → K̊, or

(2) A = (g, h)_{B} for some M-functions g, h : B → K̊, with g < h.

By inductive hypothesis, there exists an open L-cell C ⊂ K^{d} (for some d and some L ≥ M depending only on M and on n), and an L-function f : C → K^{n−1−d}, such that B = Γ(f).

In case (1) d = e. Define l : C → K^{n−e} by l(x) = ⟨f(x), g(x, f(x))⟩. It is easy to see that l is an L′-function for some L′ depending only on M and n, and that A = Γ(l).

In case (2), d = e−1. Define g̃ := g∘f, h̃ := h∘f, and B̃ := (g̃, h̃)_{C}. Given ⟨x̄, y⟩ ∈ B̃, define l(x̄, y) := f(x̄). We have that B̃ is an open e-dimensional L-cell, l : B̃ → K^{n−e} is an L-function, and A = Γ(l).

Corollary 3.11. Let A ⊆ K^{n} be definable of dimension at most e. Then there is a partition A = ⋃_{i=0}^{k} A_{i} such that dim(A_{0}) < e and A_{i} is a basic e-rectifiable set for i > 0. Moreover, the bounds of each A_{i} can be chosen to depend only on n (and not on A). We call (A_{0}, …, A_{k}) a basic e-rectifiable partition of A.

Proof. Apply Theorem 3.8 and 3.10.

Notice that a similar result has also been proved in [PW06, Theorem 2.3] (where they also take arbitrarily small bounds): however, in [PW06] they don’t require that the functions parametrizing the set A are injective (which is essential for our later uses).

4 Whitney decomposition

The fact that the functions that define an M-cell are actually Lipschitz function follows from the following property of M-cells:

Every pair of points x, y in an M-cell C ⊂ K^{n} can be connected by a definable C^{1} curve γ : [0, 1] → C with |γ′(t)| < N|x − y|, where N is a constant depending only on M and n which is finite if M is (Lemma 4.3 or [VR06] 3.10 & 3.11).

The same property implies that a N-function f on an M-cell is Lipschitz where the Lipschitz constant is finite if M and N are (Corollary 4.5). This last property will be needed for defining Hausdorff measure.

Remark 4.1. Let U ⊂ K̊^{n} be open and definable, and f : U → K̊ be an M-function (for some finite M). It is not true in general that f is L-Lipschitz for some finite L: this is the reason why we needed to prove Theorem 3.8.

Definition 4.2. Let A ⊂ K^{n}, B ⊂ K^{m} be definable sets. Let λ ⊂ A × ([0, 1] × B) ⊂ K^{n} × K^{1+m} be a definable set such that for every x ∈ A, the fiber over x

λ_{x} := {y ∈ [0, 1] × B : ⟨x, y⟩ ∈ λ}

is a curve λ_{x} : [0, 1] → B. We view λ as describing the family of curves {λ_{x}}_{x∈A}. Such a family is a definable family of curves (in B, parametrized by A).

An L-cell is an L-Lipschitz cell if the functions that define the L-cell are L-Lipschitz.

Lemma 4.3. Fix L ∈ K_{>0} and n ∈ ℕ_{>0}. Then, there is a constant K(n, L) ∈ K_{>0} depending only on n and L, that is finite if L is, such that for every L-Lipschitz cell C ⊂ K^{n} there is a definable family of curves γ ⊂ C^{2}×([0, 1]×C) such that: For all x, y ∈ C, γ_{x,y} : [0, 1] → C is a C^{1}-curve with

(i) γ_{xy}(0) = x, γ_{xy}(1) = y;

(ii) |γ′_{xy}(t)| ≤ K(n, L)|x − y|, for all t ∈ [0, 1].

Proof. By induction on n. For n = 1 the lemma is clear. Take n ≥ 1, and assume that the lemma holds for n. Let C ⊂ K^{n+1} be an L-Lipschitz cell. Then C = Γ(f) or C = (g, h)_{X} for some L-Lipschitz cell X ⊂ K^{n−1} and definable, C^{1}, L-Lipschitz functions f, g, h with g < h, and |Df|, |Dg|, |Dh| ≤ L. By induction, there are a constant k := K(n−1, L) and a definable family of C^{1}-curves β in X with the required properties. Let π_{n} : K^{n+1} → K^{n} be the projection onto the first n coordinates.

If C = Γ(f), we lift β to C via f: fix x, y ∈ C and let γ_{x,y}(t) := (α(t), f(α(t))), where for all t ∈ [0, 1] α(t) := β_{π_{n}(x),π_{n}(y)}(t). Then we have |γ′_{xy}(t)| ≤ (1 + L)k|x − y|.

If C = (g, h)_{X}, we lift β as follows: Fix x, y ∈ C and let α := β_{π_{n}(x),π_{n}(y)}. Let π : K^{n+1} → K be the projection onto the last coordinate and take u, v ∈ (0, 1) with

π(x) = uh(α(0)) + (1 − u)g(α(0))
π(y) = vh(α(1)) + (1 − v)g(α(1)).

Let l(t) := tv +(1−t)u, for t ∈ [0, 1]. We define γ_{x,y}(t) := (α(t), l(t)h(α(t))+ (1 − l(t))g(α(t))), and note that

|γ′_{xy}(t)| ≤ k|x − y| + |(v − u)(h(α(t)) − g(α(t)))| + 2Lk|x − y|,

since l(t), 1 − l(t) are between 0 and 1 and |Dh(α′(t))|, |Dg(α′(t))| ≤ L|α′(t)|. Let f := h − g. We want to bound |(v − u)f(α(t))|, which equals

|πy − πx − v(f(α(1)) − f(α(t))) + u(f(α(0)) − f(α(t))) + g(α(0)) − g(α(1))|.

But

|f(α(1)) − f(α(t))| ≤ L|α(1) − α(t)| = L|1 − t| |\frac{α(1) − α(t)}{1 − t}| ≤ L|α′(t_{0})|

for some t_{0} between t and 1. Similarly, |f(α(0)) − f(α(t))| ≤ L|α′(t_{1})|, for some t_{1} between t and 1. Since u, v ∈ [0, 1], we get

|(v − u)f(α(t))| ≤ |πy − πx| + 2Lk|x − y| + L|x − y|;

thus |γ′_{xy}(t)| ≤ K(n, L)|x − y| for some constant K(n, L) depending only on n and L which is finite if L is. The collection of the curves γ_{xy} for x, y ∈ C constitutes the required family of curves.

Theorem 4.4. Let L > 0, and let C ⊂ K^{n} be an L-cell. Then C is a k(n, L)-Lipschitz cell, where k(n, L) depends only on n and L, and is finite if L is.

Proof. By induction on n; the theorem is clear for n = 1. Assume that n > 1 and that the theorem holds for n − 1. Then C = Γ(f) or C = (g, h)_{X}, where X ⊂ K^{n−1} is a k(n − 1, L)-Lipschitz cell and f, g, h are C^{1}-functions on X such that |Df|, |Dg|, |Dh| ≤ L. We need to show that f, g, h are Lipschitz.

Since X is a k-Lipschitz cell, k := k(n − 1, L), it follows from Lemma 4.3 that there is a constant K(n − 1, k) such that whenever x, y ∈ X, there is a definable, C^{1}-curve γ joining x and y with |γ′(t)| ≤ K(n − 1, k)|x − y| for all t ∈ [0, 1]. Let g := f ∘ γ, and let t_{0} ∈ (0, 1) be such that

|f(x) − f(y)| = |g′(t_{0})| = |Df(γ′(t_{0}))| ≤ L|γ′(t_{0})| ≤ LK(n − 1, k)|x − y|.

Thus f is LK(n − 1, k)-Lipschitz. We set k(n, L) := LK(n − 1, k).

Corollary 4.5. Let C be an M-cell and f be a definable M-function. Then f is Lipschitz, and with finite Lipschitz constant if M is finite.

Proof. By Theorem 4.4, C has a definable family of curves as in Lemma 4.3. The result therefore follows from the mean value theorem.

Definition 4.6. A definable set A ⊂ K^{n} satisfies the Whitney arc property if there is a constant K ∈ K̊_{>0} such that for all x, y ∈ A there is a definable curve γ : [0, 1] → A with γ(0) = x, γ(1) = y and length(γ) := ∫_{0}^{1} |γ′| ≤ K|x − y|.

Lemma 4.7. Let C ⊂ K̊^{n} be an M-cell, M ∈ K̊. Then, C satisfies the Whitney arc property.

Proof. It follows from Theorem 4.4 and Lemma 4.3.

Theorem 4.8. Let A ⊂ K̊^{n} be definable. Then, A can be partitioned into finitely many definable sets, each of them satisfying the Whitney arc property.

Proof. This follows from Lemma 4.7, Theorem 3.8 and the fact that the Whitney arc property is invariant under an orthonormal change of coordinates.

5 Hausdorff measure

For an introduction to geometric measure theory, and in particular to the Hausdorff measure, see [Morgan88].

Definition 5.1. Let U ⊆ K^{n} be open and let f : U → K̊^{m} be a definable function. If a ∈ U, e ≤ n and M is the set of the e × e minors of Df(a) we define

J_{e}f(a) = \begin{cases}+∞ & if f is not differentiable at a or rank(Df(a)) > e, \\ \sqrt{∑_{m∈M} m^{2}} & otherwise; \end{cases}

(cf. [Morgan88, §3.6]).

Notice that if e = n = m, then J_{n}f = |det(Df)|.

Definition 5.2. Let U ⊆ K̊^{e} be an open M-cell for some M ∈ ℕ, and let f : U → K̊^{m} be a definable function with finite derivative. Let F : U → K̊^{m+e} be F (x) := ⟨x, f(x)⟩ and C := Γ(f) = F (U) (notice that C has bounded diameter). We define

H^{e}(C) := ∫_{U} J_{e}F dL^{e}.

Lemma 5.3. If C ⊆ K̊^{n} is basic e-rectifiable, then H^{e}(C) = H_{ℝ}^{e}(st(C)), where H_{ℝ}^{e} is the e-dimensional Hausdorff measure on ℝ^{n}.

Proof. Let A ⊂ K̊^{e} and f : A → K̊^{n−e} be as in Definition 3.9, and F : A → K̊^{n} as in Definition 5.2. Let B := st(A). Then, using the real Area formula [Morgan88],

∫_{A} J_{e}F dL^{e} = ∫_{B} J_{e}(\overline{F}) dL_{ℝ}^{e} = H_{ℝ}^{e}(\overline{F} (B)) = H_{ℝ}^{e}(st(C)).

Definition 5.4. Let A ⊆ K̊^{n} be definable of dimension at most e, and (A_{0}, …, A_{k}) be a basic e-rectifiable partition of A. Define

H^{e}(A) := ∑_{i=1}^{k} H^{e}(A_{i}),

where H^{e}(A_{i}) is defined using 5.2.

Lemma 5.5. If A is as in the above definition, then H^{e}(A) does not depend on the choice of the basic e-rectifiable partition (A_{0}, …, A_{k}).

Proof. It suffices to prove the following: if C is a basic e-rectifiable set and (A_{0}, …, A_{k}) is a basic e-rectifiable partition of C, then H^{e}(C) = ∑_{i=1}^{k} H^{e}(A_{i}), where H^{e}(C) and H^{e}(A_{i}) are defined using 5.2. For every i = 1, …, n let U and V_{i} be M-cells, f : U → K^{n−e} and g_{i} : V_{i} → K^{n−e} be definable functions with finite derivative, σ_{i} be a permutation of variables of K^{n}, F : K^{e} → K^{n} defined by F (x) := (x, f(x)), and G_{i} : K^{e} → K^{n} defined by G(x) = σ_{i}(x, g_{i}(x)) such that C = F (U) and A_{i} = G_{i}(V_{i}). Define U_{i} := F^{−1}(A_{i}) ⊆ U, and H_{i} := G_{i}^{−1} ∘ F : U_{i} → V_{i}. Notice that each H_{i} is a bi-Lipschitz bijection, that U is the disjoint union of the U_{i}, and that dim(U_{0}) < e. Hence,

H^{e}(C) = ∫_{U} J_{e}F dL^{e} = ∑_{i=1}^{n} ∫_{U_{i}} J_{e}F dL^{e} = ∑_{i=1}^{n} ∫_{U_{i}} J_{e}(G_{i} ∘ H_{i}) dL^{e} =

= ∑_{i=1}^{n} ∫_{U_{i}} (J_{e}(G_{i})∘H_{i})·|det(DH_{i})|dL^{e} = ∑_{i=1}^{n} ∫_{V_{i}} J_{e}G_{i}dL^{e} = ∑_{i=1}^{n} H^{e}(A_{i}),

where we used Lemma 2.9, the fact that each σ_{i} is a linear function with determinant ±1, and that J_{e}(G ∘ H) = (J_{e}(G) ∘ H) · |det(DH)|.

Lemma 5.6. H^{e} does not depend on n. That is, let m ≥ n, and A ⊂ K̊^{n} definable, and ψ : K^{n} → K^{m} be the embedding x ↦ (x, 0). Then, H^{e}(A) = H^{e}(ψ(A)).

Proof. Obvious from the definition and Lemma 5.5.

Notice that H^{0}(C) is the cardinality of C.

It is clear that H^{e} can be extended to the σ-ring generated by the definable subsets of K^{n} of finite diameter and dimension at most e; we will also denote the completion of this extension by H^{e}.

Lemma 5.7. H^{e} is a measure on the σ-ring generated by the definable subsets of K^{n} of bounded diameter and dimension at most e.

Proof. Since K is ℵ_{1}-saturated, it suffices to show that, for every A and B disjoint definable subsets of K^{n} of finite diameter and dimension at most e, H^{e}(A∪B) = H^{e}(A)+H^{e}(B). But this follows immediately from Lemma 5.5.

Example 5.8. In Lemma 5.3, the assumption that C is basic e-rectifiable is necessary. For instance, take ϵ > 0 infinitesimal, and X be the following subset of K^{2}

X := ([0, 1] × {0}) ∪ {⟨x, y⟩ : 0 ≤ x ≤ 1 & y = ϵx}.

Then, st(X) = [0, 1] × {0}, and thus H^{1}(X) = 2, while H_{ℝ}^{1}(st(X)) = 1. This is the source of complication in the theory, and one of the reasons why we had to wait until this section to introduce H^{e}.

6 Cauchy-Crofton formula

Give e ≤ n, define

β := Γ(\frac{e+1}{2})Γ(\frac{n−e+1}{2})Γ(\frac{n+1}{2})^{−1}π^{−1/2}.

Definition 6.1. Let AG_{e}(K^{n}) be the Grassmannian of affine e-dimensional subspaces of K^{n} and let AG_{e}(ℝ^{n}) be the Grassmannian of affine e-dimensional subspaces of ℝ^{n}. Fix an embedding of AG_{e}(ℝ^{n}) into some ℝ^{m}, such that AG_{e}(ℝ^{n}) is a ∅-semialgebraic closed submanifold of ℝ^{m}, and the restriction to AG_{e}(ℝ^{n}) of the dim(AG_{e}(ℝ^{n}))-dimensional Hausdorff measure coincides with the Haar measure on AG_{e}(ℝ^{n}).

Definition 6.2. Given A ⊆ K^{n} and E ∈ AG_{n−e}(K^{n}), let f_{A}(E) := #(A∩E).

Theorem 6.3 (Cauchy-Crofton Formula). Let A ⊆ K̊^{n} be definable of dimension e. Then,

H^{e}(A) = \frac{1}{β} ∫_{AG_{n−e}(K^{n})} f_{A} dL^{AG_{n−e}(K^{n})}.

We prove the theorem by reducing it to the known case of K = ℝ. This is done by showing that #(A ∩ E) equals #(st A ∩ st E) almost everywhere.

Definition 6.4. Let f : U → K̊^{m} be definable, with U ⊂ K̊^{n} open. Let E ⊂ ℝ^{n} and \overline{f} be as in Lemma 2.10. We say that b ∈ ℝ^{n} is an S-regular point of \overline{f} if

i) b ∈ st(U) \ \overline{E};

ii) b is a regular point of \overline{f}.

Otherwise, we say that b is an S-singular point and \overline{f}(b) is an S-singular value of \overline{f}. If c ∈ ℝ^{m} is not an S-singular value, we say that c is an S-regular value of \overline{f}.

Remark 6.5. Let S be the set of S-regular points of \overline{f}. Then, S is open and definable in ℝ_{K}.

Lemma 6.6 (Morse-Sard). Assume that m ≥ n. Then, the set of S-singular values of \overline{f} is L_{ℝ}^{m}-negligible,

Proof. By Lemma 2.10, E is negligible; since E is also ℝ_{K}-definable, it has empty interior and therefore dim(E) < n. Since m ≥ n, it follows that \overline{f}(E) is negligible. The set of S-singular values of \overline{f} is the union of \overline{f}(E) and the set of singular values of \overline{f}; it is therefore negligible.

Lemma 6.7 (Implicit Function). Assume that m = n. Let b ∈ ℝ^{n}. If b is an S-regular point of \overline{f} then, for every y ∈ st^{−1}(\overline{f}(b)) there exists a unique x ∈ st^{−1}(b) such that f(x) = y.

Proof. Choose x_{0} ∈ st^{−1}(b). Let A := (Df(x_{0}))^{−1}. Since b is a regular point of \overline{f}, ∥A∥ is finite. Thus we can choose r, ρ ∈ ℚ_{>0} such that B := \overline{B(b; ρ)} is contained in the set of S-regular points of \overline{f}, and

∥D\overline{f}(b′) − D\overline{f}(b)∥ < \frac{1}{2n∥A∥}, for every b′ ∈ B
r ≤ \frac{ρ}{2∥A∥}.

Moreover, we can pick ρ such that B′ := \overline{B(x_{0}; ρ)} ⊂ U. Given y ∈ K^{n} such that |y − f(x_{0})| < r, consider the mapping

\begin{aligned}T_{y} & : B′ → K^{n} \\ T_{y}(x) & := x + A · (y − f(x)). \end{aligned}

T_{y} is definable and Lipschitz, with Lipschitz constant 1/2. Therefore, for every y ∈ B(f(x_{0}); r) there exists a unique x ∈ B′ such that T_{y}(x) = x. Thus, there is a unique x ∈ B with f(x) = y. It remains to show that, given y ∈ st^{−1}(\overline{f}(b)) and x ∈ B′ such that f(x) = y, we have x ∈ st^{−1}(b). We can verify that

\begin{aligned}\overline{T}_{y} & : B → B \\ \overline{T}_{y}(b′) & = b′ + (D\overline{f}(b))^{−1} · (\overline{f}(b) − \overline{f}(b′)) \end{aligned}

is also a contraction, and therefore it has a unique fixed point, namely b. Since \overline{T}_{y}(st(x)) = st(x), we must have st(x) = b.

Remark 6.8. Let U ⊂ K̊^{m}. If f : U → K̊^{n} is definable and M-Lipschitz (for some finite M), n ≥ m and E is L_{ℝ}^{m}-negligible, then the set f(st^{−1}(E)) is L^{n}-negligible.

Proof. We can cover E with a polyrectangle Y whose measure is an arbitrarily small rational number λ and such that Y covers st^{−1}(E). Since f(Y) has measure at most CM^{n}λ (C depends only on m and n) the result follows.

Lemma 6.9. Let A ⊆ K̊^{n} be a basic e-rectifiable set of dimension e. Consider V := K^{e} as embedded in K^{n} via the map x ↦ ⟨x, 0⟩. Identify each p ∈ V with the (n − e)-dimensional affine space which is orthogonal to V and intersects V in p. Then, for almost every p ∈ V, we have #(p ∩ A) = #(st(p) ∩ st(A)).

Proof. Let π : K^{n} → V be the orthogonal projection. Let U ⊂ K̊^{e} be an open M-cell and f : U → K^{n−e} be a definable M-function (M finite) such that A = Γ(f). Let F (x) := ⟨x, f(x)⟩. Let h := π ∘ F : U → V, and consider \overline{h} : C → st(V), C ⊂ st(U) as in Lemma 2.10. For almost every p ∈ V, #(p ∩ A) = #(h^{−1}(p)), and #(st p ∩ stA) = #(\overline{h}^{−1}(st p)) because F : U → A and \overline{F} : C → Im(\overline{F}) are bijections. Thus, it suffices to prove that, for almost every p ∈ V, #(h^{−1}(p)) = #(\overline{h}^{−1}(st p)). Let E be as in Lemma 2.10. By Remark 6.8, h(st^{−1}(E)) is L^{e}-negligible. Let S be the set of S-singular values of \overline{h}, by Lemma 6.6, S is negligible.

Let p ∈ V \ (st^{−1}(S) ∪ h(st^{−1}(E)). Then for every x in h^{−1}(p), st(x) is an S-regular point of \overline{h}, and therefore Lemma 6.7 implies #(h^{−1}(p)) = #(\overline{h}^{−1}(st p)).

Notice that the above lemma does not hold if A is only definable, instead of basic e-rectifiable.

Proof of Theorem 6.3. By Corollary 3.11, w.l.o.g. A is basic e-rectifiable. Let B := st(A), and f_{B}(F) := #(B∩F), for every F ∈ AG_{e}(ℝ^{n}). By Lemma 6.9,

∫_{AG_{n−e}(K^{n})} f_{A} dL^{AG_{n−e}(K^{n})} = ∫_{AG_{n−e}(ℝ^{n})} f_{B} dL^{AG_{n−e}(ℝ^{n})}.

By the usual Cauchy-Crofton formula [Morgan88, 3.16], the right-hand side in the above identity is equal to H_{ℝ}^{e}(B) = H^{e}(A), where we applied Lemma 5.3.

7 Further properties of Hausdorff measure and the Co-area formula

Theorem 7.1. Let e ≤ n and C ⊆ K^{n} be bounded and definable of dimension at most e.

1. H^{e} is invariant under isometries.

2. For every r ∈ K̊, H^{e}(rC) = st(r)^{e}H^{e}(C).

3. If C is ∅-semialgebraic, then H^{e}(C) = H^{e}(C_{ℝ}) = H^{e}(st(C)).

4. if dim(C) < e, then H^{e}(C) = 0; the converse is not true.

5. H^{e}(C) < +∞.

6. If (C(r))_{r∈K^{d}} is a definable family of bounded subsets of K^{n}, then there exists a natural number M such that H^{n}(C(r)) < M for every r ∈ K^{d}.

7. If K′ is either an elementary extension or an o-minimal expansion of K, then H^{e}(C_{K′}) = H^{e}(C).

8. If n = e, then H^{e}(C) = L^{n}(C).

9. If C is a subset of an e-dimensional affine space E, then H^{e}(C) =

L^{E}(C). Proof.

(1) Use the Cauchy-Crofton formula.

(2), (4) and (7) Apply the definition of H^{e} and Lemma 5.5.

(3) Apply Corollary 3.11 to C_{ℝ} and use Lemma 5.3.

(5) and (6) Apply the Cauchy-Crofton formula: see [Dries03]. (8) Apply Lemma 5.3.

(9) Since H^{e} is invariant under isometries, w.l.o.g. E is the coordinate space K^{e}. By Lemma 5.6, the measure H^{e} inside K^{n} is equal to the measure H^{e} inside K^{e}, and the latter is equal to L^{e}. The conclusion follows from Remark 2.1.

The following theorem is the adaption to o-minimal structures of the Coarea formula, a well-known generalization of Fubini’s theorem. Let D := [0, 1] ⊂ K. Theorem 7.2 (Co-area Formula). Let A ⊂ D^{m} be definable, and f : D^{m} → D^{n} be a definable Lipschitz function, with m ≥ n. Then, J_{n}f is L_{K}^{m}- integrable, and

∫_{A} J_{n}f dL^{m} = ∫_{D^{n}} H^{m−n}(A ∩ f^{−1}(y)) dL^{n}(y).

Sketch of Proof. W.l.o.g., A is an open subset of D^{m}. By Lemma 6.6, w.l.o.g. all points of A are S-regular for \overline{f}. Apply the real co-area formula [Morgan88] to g := \overline{f} and B := st(A), and obtain

∫_{A} J_{n}f dL^{m} = ∫_{B} J_{n}g dL_{ℝ}^{m} = ∫_{D_{ℝ}^{n}} H_{ℝ}^{m−n}(B ∩ g^{−1}(z)) dL_{ℝ}^{n}(z).

By the Implicit Function Theorem and Lemma 5.3, for almost every y ∈ D_{ℝ}^{n}, we have

H^{m−n}(A ∩ f^{−1}(y)) = H_{ℝ}^{m−n}(B ∩ g^{−1}(st y)).

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