Definability in affine continuous logic

Abstract: I study definable sets in affine continuous logic. Let $T$ be an affine theory. After giving some general results, it is proved that if $T$ has a first order model, its extremal theory is a complete first order theory and first order definable sets are affinely definable. In this case, the type spaces of $T$ are Bauer simplices and they coincide with the sets of Keisler measures of the extremal theory. In contrast, if $T$ has a compact model, definable sets are exactly the end-sets of definable predicates. As an example, it is proved in the theory of probability algebras that one dimensional definable sets are exactly the intervals $[a,b]$.