One of the central problems in the study of von Neumann algebras is to find computable invariants which can distinguish nonisomorphic algebras. In the 1980s, Dan Voiculescu developed a noncommutative probability theory in order to understand a particular class of such von Neumann algebras. Specifically, he defined subsets of $R^n$ called microstate spaces which model the behavior of a generating set of a given von Neumann algebra. Since these spaces are subsets of $R^n$, classical analytic tools such as volume can be applied to them. I will discuss how the application of ideas from geometric measure theory to microstate spaces has provided insight into the general problem of invariants and answered some longstanding questions in von Neumann algebras.