The “nice singularities”: O-minimal geometry

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The HKP Integral on integral currents of dimension 1.

Definition 2.3.4. A function f defined kTk almost everywhere on set1 kTk, is Pfeffer 1 integrable or HKP integrable on T if there exists a continuous additive function F on S6(T) and for every  » > 0, there exists a gauge and a positive number such that whenever P is a -fine tagged family in T with jF(T 􀀀 [P])j < , there holds: jF(T) 􀀀 (f; P)j < « : (2.6) (Where (f; P) denotes the Riemann sum P (x;S)2P f(x)M(S).) F(T) is also the HKP integral of f on T and we sometimes denote it (HKP) R T f. Question 2.3.5. Is it equivalent to ask that each families be surbordinate to some decomposition? This is not clear because a piece of T can very well not be a piece of any decomposition (see figure 2.2).
According to example 2.1.13, it is not sufficient to be integrable on all elements of one given decomposition to be integrable on the whole current. However, suppose f is integrable on each piece for two decompositions, is the integral the same?
We list the main basic properties of the integral, when the proofs are not given, they are similar to the ones in chapter 5 section 5.1 and the first section of this chapter: Proposition 2.3.6 (Space of Pfeffer 1 integrable functions on T). The space of Pfeffer 1 integrable functions on T is a linear space and the integral: f 7! I(f; T) is a linear operator. Furthermore, if f 6 g and f and g are HKP integrable on T, then (HKP) R T f 6 (HKP) R T g.

Fundamental Theorem of Integration

Proposition 2.3.17. If F is a continuous additive function on S6(T) which is AC and derivable kTk almost everywhere, then x 7! DT F(x) is HKP integrable on T with indefinite integral F. Proof. Let N be the set of non derivability points of F in set1 kTk. Let f be the function defined on set1 kTk by f(x) = 0 if x 2 N and f(x) = DT F(x) otherwise. For  » > 0, let be a gauge on set1 kTk such that whenever P is a -fine tagged family in T anchored in N, jF([P])j <  » and for all x 2 set1 kTknN, (x) is a positive number such that for all S 2 S6(T; x; (x)) jF(S) 􀀀 f(x)M(S)j < « M(S).

Topology on the space of subcurrents of T

Recall that in [19], the authors define a topology FX;m on the space Nm(X) of normal currents of dimension m supported in a set X Rn. This topology is not metrizable, though it is hereditarily sequential and it coincides with the flat norm topology on every space of the form: Nm;c(X) := Nm(X) \ fS;N(S) < cg: Consider the space S(T) of subcurrents of a given integral current T 2 Im(Rn). Clearly all the currents S 2 S(T) are in Nm(spt T).
Theorem 3.1.6. For c > 0, the space S(T) \ Nm;c(spt T) is compact in (Nm(spt T); Fspt T;m). Furthermore, the mass operator M is continuous on S(T) and convergence in the flat norm is equivalent to convergence in mass. Since Nm(spt T) is sequential and spt T is compact, for the first claim, it suffices to prove that any sequence in S(T) with bounded normal mass has a subsequence which converges in S(T).

The special case of codimension 0: sets of finite perimeter

Some definitions and classical results on sets of finite perimeter are contained in Appendix C. In this paragraph, we outline the main properties of currents representing sets of finite perimeter which are not shared by general integral currents.
Let us first point out that the isoperimetric inequality doesn’t hold for general integral currents, as can be shown by considering the two dimensional currents associated to spheres of arbitrary diameter in R3. This example also shows that the flat norm and the mass norm are not equivalent in positive codimension.

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Gauges and families in a set of finite perimeter

A function : A Rm ! [ 0;+1) is called a gauge in A if its zero set 􀀀1(0) A is thin , that is, has -finite Hm􀀀1 measure. A family in A is a finite collection of non overlapping sets of finite perimeter B1; : : : ;Bp, where each Bj is contained in A and jBj \ Bj0 j = 0 whenever j 6= j0, together with tags x1; : : : ; xp, where xj 2 cle A \ clBj for all j. If is a gauge on cle A, a family ((B1; x1); : : : ; (Bp; xp)) in A is -fine whenever diam(Bj) < (xj) for all j. Given a positive , a bounded set of finite perimeter B is called -regular if jBj kBk diamB > 0.

Table of contents :

1 Introduction 
1.1 Fundamental Theorems and singularities
1.1.1 Fundamental Theorem of Calculus
1.1.2 The Divergence Theorem and Pfeffer integration
1.1.3 Stokes’ Theorem and integral currents
1.1.4 The “nice singularities”: O-minimal geometry
1.2 Summary of the thesis
1.3 Applications and perspectives
2 One dimensional integration 
2.1 The integral of Kurzweil and Henstock
2.1.1 Definition and classical properties
2.1.2 AC functions and the Fundamental Theorem of HK Integration
2.1.3 An equivalent definition of the HK integral
2.2 Current of dimension 1, decompositions, pieces
2.2.1 Subcurrents, pieces of current
2.2.2 Continuous function on the space of pieces of T
2.2.3 Derivation
2.3 Integration on currents of dimension 1
2.3.1 Howard Cousin Lemma in dimension 1
2.3.2 AC functions on S6(T)
2.3.3 The HKP Integral on integral currents of dimension 1
2.3.4 Fundamental Theorem of Integration
3 Subcurrents and derivation. 
3.1 Currents and subcurrents
3.1.1 Subcurrents: definition and properties
3.1.2 Topology on the space of subcurrents of T
3.1.3 The special case of codimension 0: sets of finite perimeter
3.1.4 Intersection of subcurrents
3.1.5 Inner approximation of currents
3.2 Functions on the space of subcurrents
3.2.1 Continuous functions on S(T), m-charges and flat cochains
3.2.2 Derivation of a function on a current
4 Howard-Cousin Property and Stokes’ Theorem 
4.1 The Howard-Cousin Lemma
4.1.1 Cubes
4.1.2 Continuous functions on sets of finite perimeter
4.1.3 Gauges and families in a set of finite perimeter
4.1.4 Statement and sketch of proof
4.1.5 Proof of Howard-Cousin Lemma
4.2 Howard-Cousin Property for currents
4.2.1 The property
4.2.2 Disposable sets for a current
4.2.3 A criterion to have the Howard-Cousin Property
4.2.4 Minkowski content and disposability
4.2.5 Mass minimizing currents and stationary varifolds
4.3 Stokes-Cartan Theorem for the Lebesgue Integral
4.3.1 Outline and statement
4.3.2 Proof
4.4 Counter-examples
4.4.1 The main example
4.4.2 A variation with only one singular point
4.4.3 An example related to the hereditary Howard-Cousin Property
4.4.4 Remarks on other counter-examples
5 Integration on currents 
5.1 Pfeffer Integration
5.2 Stokes’ Theorem for Pfeffer Integration
6 Definable chains in o-minimal structures 
6.1 O-minimality
6.2 Properties of definable sets, definable families
6.2.1 Cell decomposition
6.2.2 Control of Hausdorff measures
6.3 Definable chains
Appendix
Appendix A Measure Theory
Appendix B Integral currents
B.1 Notations and classical definitions
B.2 Currents
B.3 Slicing
B.4 Facts on integral currents of dimension 1
Appendix C Sets of finite perimeter

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