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In the classification theory of first order theories, various concepts of rank, dimension, and measure are used to study the complexity of the first-order definable subsets of models of the theories. Macpherson and Steinhorn [1] introduced a dimension-measure pair for the definable subsets of finite structures. This was generalized by Elwes and Macpherson [2] to ultraproducts of "asymptotic classes"; this includes pseudofinite structures.

Charlotte Kestner, in her PhD thesis and in [3], describes the nature of the Macpherson-Steinhorn measure for theories of modules. An MS-measurable module is superstable. Kestner shows that it follows from the well-known pp-elimination of quantifiers for theories of modules that the MS-measurability of a theory of modules depends only on the properties of the definable-over-the-empty set subgroups.

1. One-dimensional asymptotic classes of finite structures. Trans. Am. Math. Soc. 360(1), 411-448 (2008).
2. A survey of asymptotic classes and measurable structures. In: Model Theory with applications to Algebra and Analysis. Vol. 2, Vol 350 of Londan Math. Soc. Lecture Note Ser., pp125-159, Cambridge University Press, (2008)
3. Measurability in Modules. Arch. Math. Logic (2014) 53:593-620.