There is a long history of interaction between probability and logic.  One such context has been the study of countable structures whose relations and functions are assigned probabilities rather than binary truth values; of particular interest are those assignments that do not depend on the labeling of the underlying set.  In this talk I will discuss the early work of Gaifman, Scott and Krauss that introduced this line of investigation, and then describe an equivalent setting from descriptive set theory, consisting of a certain space of countable structures with fixed underlying set.  The ergodic probability measures on this space that are invariant under permutations of the underlying set provide a natural notion of exchangeable model-theoretic structure: to each such measure there is associated a complete and consistent infinitary theory.  Furthermore, any such measure can be seen as a limit of sampling measures derived from a convergent sequence of finite structures.  I will discuss these connections, and present some related new results due to Ackerman, Freer, Kruckman and myself.