  We study the complexity of isomorphism of classes of metric structures using
methods from infinitary continuous logic. For Borel classes of locally compact
structures, we prove that if the equivalence relation of isomorphism is
potentially $\mathbf{\Sigma}^0_2$, then it is essentially countable. We also
provide an equivalent model-theoretic condition that is easy to check in
practice. This theorem is a common generalization of a result of Hjorth about
pseudo-connected metric spaces and a result of Hjorth--Kechris about discrete
structures. As a different application, we also give a new proof of Kechris's
theorem that orbit equivalence relations of actions of Polish locally compact
groups are essentially countable.
