I don't know much about it, I'm afraid. But there is the very cool fact that the usual Riemann integral can be written as basically just a sum in nonstandard analysis; it's covered very neatly in André Pétry, "Analyse Infinit́esimale: une présentation non standard" which is very readable. There's a lot more advanced stuff, including Brownian motion and (I think) some integration, in Hurd and Loeb, "An Introduction to Nonstandard Real Analysis".

For the foundations of nonstandard analysis, and a wide-ranging overview, I strongly recommend Robert Goldblatt, "Lectures on the Hyperreals: an Introduction to Nonstandard Analysis", one of the Graduate Texts in Mathematics.

I'm rusty, but did think about this a bit at one point. I recall you can't really do the usual constructions on hyperreals (problems with completeness?) but you can do something like approximate any real-valued measure closely.