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](https://web.math.princeton.edu/~nelson/books/rept.pdf)
* [Geyer (2007). Radically Elementary Probability and Statistics](http://www.stat.umn.edu/geyer/nsa/o.pdf)


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?



Nonstandard analysis in probability theory
Non-standard analysis has been quite successful in settling existence questions in probability theory. [Hyperfinite Loeb spaces](http://www.jstor.org/stable/1997222) allow for several constructions that cannot be done on standard probability spaces. In particular, NSA was quite useful for the construction of certain adapted processes. There is a paper by Hoover and Keisler, [Adapted Probability Distributions](http://www.jstor.org/stable/1999401?&Search=yes&searchText=Adapted&searchText=probability&searchText=distributions&list=hide&searchUri=%252Faction%252FdoBasicSearch%253Ffilter%253Djid%25253A10.2307%25252Fj100071%2526Query%253DAdapted%252Bprobability%252Bdistributions%2526Search.x%253D0%2526Search.y%253D0%2526wc%253Don&prevSearch=&item=1&ttl=60&returnArticleService=showFullText), from 1984, in which the authors show that many of the properties that make hyperfinite Loeb spaces so useful where due to a property they called saturation: A probability space $(\Omega,\Sigma,\mu)$ is *saturated* if whenever $\nu$ is a Borel probability measure on $[0,1]^2$ and $f:\omega\to[0,1]$ a random variable with distribution equal to the marginal of $\nu$ on the first coordinate, then there exists a random variable $g:\Omega\to[0,1]$ such that the distribution of $(f,g)$ is $\nu$. An example of a saturated probability space that is not a hyperfinite Loeb space is the coin-flipping measure on $\{0,1\}^\kappa$ when $\kappa$ is uncountable. A relatively readable exposition of this approach can be found in the small book [Model Theory of Stochastic Processes by Fajardo and Keisler](http://rads.stackoverflow.com/amzn/click/1568811721). There are also several related papers and surveys [on Keisler's homepage](http:///).


In a sense, we nowadays understand fairly well how certain powerful techniques of non-standard analysis work below the surface, so we can use a lot of the constructions freed of NSA. There isn't really anything where it is *necessary* to use NSA. Still, NSA is a rather powerful and useful tool. A good overview over what it can do for probability theory, mainly the theory of sochastic processes, is in the article by Osswald and Sun in [Nonstandard Analysis for the Working Mathematician](http://rads.stackoverflow.com/amzn/click/079236340X) by Loeb.


