A saturation property of structures obtained by forcing with a compact family of random variables

A method for constructing Boolean-valued models of some fragments of arithmetic was developed in Krají?ek (Forcing with Random Variables and Proof Complexity, London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge, 2011), with the intended applications in bounded arithmetic and proof complexity. Such a model is formed by a family of random variables defined on a pseudo-finite sample space. We show that under a fairly natural condition on the family called compactness in Krají?ek (Forcing with Random Variables and Proof Complexity, London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge, 2011)] the resulting structure has a property that is naturally interpreted as saturation for existential types. We also give an example showing that this cannot be extended to universal types.