##
**Derivation and integration.**
*(English)*
Zbl 0980.26008

Cambridge Tracts in Mathematics. 140. Cambridge: Cambridge University Press. xvi, 266 p. (2001).

The problem of integrating derivatives was completely solved for functions of a real variable with the Denjoy, Perron and Henstock-Kurzweil integrals. However, extensions of these integrals to Euclidean \(n\)-space have proved to be of little value, lacking such basic requirements as invariance under rotations. The current work is the author’s solution to the problem of recovering a function from its derivative for functions on \({\mathbb R}^m\). In this he is to be praised for successfully solving a long outstanding problem and for presenting his work in an admirable manner. The work builds on progress made by the author and other writers (principally B. Bongiorno, Z. Buczolich and B. S. Thomson). This book completely supersedes Part II of the author’s earlier work, [The Riemann approach to integration: local geometric theory’. Cambridge: Cambridge University Press (1993; Zbl 0804.26005)].

The key concept is that of a charge, which is a finitely additive set function defined on BV sets in \({\mathbb R}^m\). (A set is BV if it has a (De Giorgi) BV characteristic function.) A type of continuity is defined \((\text{AC}_*)\) and the \(\text{AC}_*\) charges are the multi-dimensional analogues of the \(\text{ACG}_*\) functions of the Denjoy theory.

Derivates of charges are defined with respect to Lebesgue measure. A derivation base is made up of BV sets with a suitable regularity. (The regularity of a bounded BV set is the ratio of its Lebesgue measure to the product of its perimeter and diameter.) The critical variation of a charge is defined in terms of a certain associated Borel regular measure and it is proved that a charge is absolutely continuous if and only if its critical variation is absolutely continuous and locally finite.

Both, Denjoy and Henstock-Kurzweil style integration processes are defined and it is shown that an \(\text{AC}_*\) charge is derivable almost everywhere and can be recovered from its derivate, even when the derivate is not Lebesgue integrable. The resulting integral is invariant under lipeomorphisms (bijective Lipschitz maps with a Lipschitz inverse) and a theorem on change of variables under lipeomorphisms is developed. This leads to a very general version of the Stokes theorem. Another result is the characterisation of multipliers. Further results and generalisations also appear.

The writing style is clear and straightforward. There are numerous complex proofs but the work has been laid out in a logical manner. Various observations, examples and remarks help to elucidate the material. Some open problems are presented but no exercises. Although a chapter on background material is provided, the reader is expected to have a working knowledge of measure theory (including Hausdorff measure). Previous exposure to BV sets would be helpful. The book is well referenced and the bibliography contains 88 items. A useful list of symbols and an index are included. The binding is of reasonable quality.

The key concept is that of a charge, which is a finitely additive set function defined on BV sets in \({\mathbb R}^m\). (A set is BV if it has a (De Giorgi) BV characteristic function.) A type of continuity is defined \((\text{AC}_*)\) and the \(\text{AC}_*\) charges are the multi-dimensional analogues of the \(\text{ACG}_*\) functions of the Denjoy theory.

Derivates of charges are defined with respect to Lebesgue measure. A derivation base is made up of BV sets with a suitable regularity. (The regularity of a bounded BV set is the ratio of its Lebesgue measure to the product of its perimeter and diameter.) The critical variation of a charge is defined in terms of a certain associated Borel regular measure and it is proved that a charge is absolutely continuous if and only if its critical variation is absolutely continuous and locally finite.

Both, Denjoy and Henstock-Kurzweil style integration processes are defined and it is shown that an \(\text{AC}_*\) charge is derivable almost everywhere and can be recovered from its derivate, even when the derivate is not Lebesgue integrable. The resulting integral is invariant under lipeomorphisms (bijective Lipschitz maps with a Lipschitz inverse) and a theorem on change of variables under lipeomorphisms is developed. This leads to a very general version of the Stokes theorem. Another result is the characterisation of multipliers. Further results and generalisations also appear.

The writing style is clear and straightforward. There are numerous complex proofs but the work has been laid out in a logical manner. Various observations, examples and remarks help to elucidate the material. Some open problems are presented but no exercises. Although a chapter on background material is provided, the reader is expected to have a working knowledge of measure theory (including Hausdorff measure). Previous exposure to BV sets would be helpful. The book is well referenced and the bibliography contains 88 items. A useful list of symbols and an index are included. The binding is of reasonable quality.

Reviewer: Erik Talvila (Edmonton)

### MSC:

26B15 | Integration of real functions of several variables: length, area, volume |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26B20 | Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.) |

26A39 | Denjoy and Perron integrals, other special integrals |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |