The Keisler Order in Continuous Logic

We extend the Keisler order to continuous first-order theories. In the process, we show that if F is a λ-regular filter on I , and Mi i∈I , Ni i∈I are sequences of continuous structures in the same language such that ΠF Mi and ΠF Ni have the same continuous first-order theory, then the classical structures corresponding to ΠF Mi and ΠF Ni satisfy the same sentences of L∞,λ + of alternating quantifier rank at most (λ).