Integration on measure chains. (English) Zbl 1083.26005

Aulbach, Bernd (ed.) et al., New progress in difference equations. Proceedings of the 6th international conference on difference equations, Augsburg, Germany July 30–August 3, 2001. Boca Raton, FL: CRC Press (ISBN 0-415-31675-8/hbk). 239-252 (2004).
Summary: In its original form the calculus on measure chains is mainly a differential calculus. The notion of integral being used, the so-called Cauchy integral, is defined by means of antiderivatives and, therefore, it is too narrow for the development of a full infinitesimal calculus.
In this paper, we present several other notions of integral such as the Riemann, the Cauchy-Riemann, the Borel and the Lebesgue integral for functions from a measure chain to an arbitrary real or complex Banach space. As in ordinary calculus, of those notions only the Lebesgue integral provides a concept which ensures the extension of the original calculus on measure chains to a full infinitesimal calculus including powerful convergence results and complete function spaces.
For the entire collection see [Zbl 1052.39001].


26A42 Integrals of Riemann, Stieltjes and Lebesgue type
28A25 Integration with respect to measures and other set functions
28B05 Vector-valued set functions, measures and integrals
39A10 Additive difference equations
39A12 Discrete version of topics in analysis