TITLE: Nonstandard analysis in probability theory
QUESTION [36 upvotes]: I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books: 

Nelson (1987). Radically Elementary Probability Theory
Geyer (2007). Radically Elementary Probability and Statistics

Although Nelson's book is several decades old, as far as I can see, its approach has not yet caught on. Also, I couldn't find a lot of papers published in the leading probability journals on that topic. I am quite intrigued by that phenomenon. My questions are the following

Why hasn't nonstandard analysis been widely adopted by probabilists?
Were there some success stories in some particular sub-fields of probability theory or statistics?
Does there exist some known fundamental objections in probability theory to the approach in there?

REPLY [2 votes]: Probabilists are no different than the conventional analyst, at least in most places I have been, Eastern Europe (my home), and US. NSA has been seen a lot more as an alternative rather than competitor. Model theorists are more interested in studying different models than actually electing one best and propose a universal transition. The work of Nelson has been revisited. Besides the work of Geyer in your post there is a terrific recent book by Herzberg and articles by Weisshaupt (Journal of Logic and Analysis 2009 and 2011) and Andrade (Positivity, doi 10.1007/s11117-015-0333-9) and also an article by Geyer and Andrade (Journal of Logic and Analysis).
I dont agree with other post saying that Nelson's NSA is more of interest to non mathematicians. Nelson, Herzberg and Weisshaupt are mathematicians, Geyer has degree in Physics but is now statistician as is Andrade (also see work on NS Brownian motion in Physica A, 2015 Volume 429). I say interest is also high in physics and mathematical finance as illustrated by Herzberg's book as in pure maths and stats