Cut and project sets are well-studied models of almost-periodic discrete subsets of $R^d$. In 2014 Marklof and Strombergsson introduced a natural class of random processes which generate cut and project sets in a way which is invariant under the group $ASL(d,R)$. Using Ratner's theorem and the theory of algebraic groups we classify all these measures. Using the classification we obtain results analogous to those of Siegel, Rogers, and Schmidt in geometry of numbers: summation formulas and counting points in large sets for typical cut and project sets. Joint work with Rene Ruehr and Yotam Smilansky.