##
**The integral as a limit of integral sums.**
*(English)*
Zbl 0554.26007

Jahrb. Überblicke Math. 1984, Math. Surv. 17, 105-136 (1984).

[For the entire collection see Zbl 0535.00005.]

This paper is an excellent survey on the Riemann sum approach to Perron- Ward-like integrals introduced by Kurzweil in 1957 and independently by Henstock in 1961. Such integrals are usually referred as S-integrals, Riemann-complete integrals or generalized integrals. The basic theory is recalled, as well as the relation with the Lebesgue integral, in particular following McShane’s approach. The classical convergence and Fubini theorems are then discussed, as well as the integration by parts and the multiplication of S-integrable functions. Some generalizations are then described, and in particular the ones which lead to Stokes-type theorems for mere differentiable vector fields and transformation formulas. Finally, the connection of the S-integral with ordinary differential equation is sketched. The paper is a very readable introduction to this simple but fruitful approach to various aspects of integration theory and the reader can continue his studies in this direction by consulting the references given at the end of the paper.

This paper is an excellent survey on the Riemann sum approach to Perron- Ward-like integrals introduced by Kurzweil in 1957 and independently by Henstock in 1961. Such integrals are usually referred as S-integrals, Riemann-complete integrals or generalized integrals. The basic theory is recalled, as well as the relation with the Lebesgue integral, in particular following McShane’s approach. The classical convergence and Fubini theorems are then discussed, as well as the integration by parts and the multiplication of S-integrable functions. Some generalizations are then described, and in particular the ones which lead to Stokes-type theorems for mere differentiable vector fields and transformation formulas. Finally, the connection of the S-integral with ordinary differential equation is sketched. The paper is a very readable introduction to this simple but fruitful approach to various aspects of integration theory and the reader can continue his studies in this direction by consulting the references given at the end of the paper.

Reviewer: J.Mawhin

### MSC:

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

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

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

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

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