These notes present the theory of generalized Riemann integral, due to R. Henstock and J. Kurzweil, from a nonstandard point of view.
The key notion we use, that of a-ultrasmall numbers, is due to B. Benninghofen and M. M. Richter, A general theory of superinfinitesimals, who call them "superinfinitesimals." E. Gordon, in Nonstandard Methods in Commutative Harmonic Analysis, developed an approach to relative standardness that is different from that of Y. Péraire; in particular, his relative infinitesimals are the superinfinitesimals. Here we have combined the two techniques.
B. Benninghofen presented an approach to the generalized Riemann integral using superinfinitesimals in Superinfinitesimals and the calculus of the generalized Riemann integral, in Models and Sets, G. H. Müller and M. M. Richter,Our development of the generalized Riemann integral follows the excellent exposition in R. Bartle; A Modern Theory of Integration, to which the reader is referred for further study of this topic.