In joint work with Haosui Duanmu and William Weiss, we investigate applications of nonstandard analysis in measure theory. The use of infinitesimals and hyperfinite probability spaces offers alternative viewpoints on many classical problems, via Peter Loeb's construction of measures using hyperfinite probability mass functions to construct classical measures. In this talk, I will describe a solution to a problem posed by Keisler, about Loeb measures, and also mention applications in statistics which are joint work with Haosui Duanmu and Daniel Roy.