Hello dear StackExchange,
in my upcoming bachelor's thesis, I plan to present an overview of some topics in nonstandard analysis, including some higher applications, like Loeb Measures (among other things). This turns out to be more confusing to realise than I hoped.
Now since this is just a bachelor's thesis, space is limited and 40 - 50 pages would already be rather long. Therefore, constructing the hyperreals from scratch via the ultrapower construction and then developing Loeb Measures is simply too much.
(although I'm not opposed to working through this to gain a deeper understanding of the topics at hand; fortunately, I learned the necessary background knowledge in my logic course)
Naturally, the next best option is an axiomatic approach, which is generally a good choice since my professor is probably more interested in direct application of NSA rather than all the machinery behind it.
Fortunately, there are already several axiomatic treatments of NSA, presenting conservative extensions of ZFC, for example Nelson's internal set theory and Hrbaceks external set theories - but regarding the construction of Loeb Measures, I have only found resources using the ultrapower construction beforehand.
So finally, my question is if there are any resources (if it's possible) where the axiomatic approach is used to shortcut all the logic and set theory, so that mathematicians unschooled in those topics can apply NSA methods to (for example) probability, using Loeb Measures.
Also, I am a bit overwhelmed by the number of different approaches with different pros and cons, and which one to choose, so any help is appreciated.
Thanks in advance!