We define an ultraproduct of metric structures based on a maximal probability charge and prove a variant of Łoś theorem for linear metric formulas. We also consider iterated ultraproducts.

Linear continuous logic is the fragment of continuous logic obtained by restricting connectives to addition and scalar multiplications. Most results in the full continuous logic have a counterpart in this fragment. In particular a linear form of the compactness theorem holds. We prove this variant and use it to deduce some basic preservation theorems.

CONTINUOUS MODEL THEORY CHAPTER I TOPOLOGICAL PRELIMINARIES. Notation Throughout the monograph our mathematical notation does not differ drastically from ...

We show that type-definable subgroups of measure algebras are definable.