##
**The Riemann approach to integration: local geometric theory.**
*(English)*
Zbl 0804.26005

Cambridge Tracts in Mathematics. 109. Cambridge: Cambridge University Press. xiv, 302 p. (1993).

Traditionally, the divergence theorem or Gauss’ theorem was proved for continuously differentiable vector fields. Various extensions have been made by Mawhin, Pfeffer, Kurzweil, and JarnĂk. The present book gives a clear exposition of an account of the extension to discontinuously differentiable vector fields using the Henstock-Kurzweil integral.

The book is divided into two parts: one-dimensional and multidimensional. In each part, the McShane integral (an absolute version of the Henstock- Kurzweil integral) is studied in detail first. Nine versions of the divergence theorem are given in the book. In order to state the theorem in general form, the concepts of gages and calibers are introduced in Section 6.7 for the one-dimensional case in anticipation of its extension to the multidimensional one later. Roughly speaking, a gage is a positive function except for a thin set, e.g., a countable set; a caliber is a sequence of positive numbers which is used to control the size of the figures (elementary sets) in a partition. Then the gage integral is defined using Riemann sums which depend on a gage and a caliber. A partial descriptive definition is given for the one-dimensional Henstock- Kurzweil integral only, using \(\text{AC}_ *\) functions (strong Lusin condition). An example (Example 11.1.2) is also given to show that no Fubini’s theorem can be proved for the gage integral as defined in the book. The last two chapters are devoted to further developments, including the F-integral and the BV-integral. However, the multipliers problem (Remark 12.8.6) remains open.

The book is divided into two parts: one-dimensional and multidimensional. In each part, the McShane integral (an absolute version of the Henstock- Kurzweil integral) is studied in detail first. Nine versions of the divergence theorem are given in the book. In order to state the theorem in general form, the concepts of gages and calibers are introduced in Section 6.7 for the one-dimensional case in anticipation of its extension to the multidimensional one later. Roughly speaking, a gage is a positive function except for a thin set, e.g., a countable set; a caliber is a sequence of positive numbers which is used to control the size of the figures (elementary sets) in a partition. Then the gage integral is defined using Riemann sums which depend on a gage and a caliber. A partial descriptive definition is given for the one-dimensional Henstock- Kurzweil integral only, using \(\text{AC}_ *\) functions (strong Lusin condition). An example (Example 11.1.2) is also given to show that no Fubini’s theorem can be proved for the gage integral as defined in the book. The last two chapters are devoted to further developments, including the F-integral and the BV-integral. However, the multipliers problem (Remark 12.8.6) remains open.

Reviewer: Lee Peng-Yee (Singapore)

### MSC:

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

26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |

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

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