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,