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We show that a version of López-Escobar’s theorem holds in the setting of model theory for metric structures. More precisely, let denote the Urysohn sphere and let Mod(,) be the space of metric -structures supported on . Then for any Iso()-invariant Borel function f: Mod(,) → [0,1], there exists a sentence ϕ of ${}_{\omega ₁\omega }$ such that for all M ∈ Mod(,) we have $f\left(M\right)={\varphi }^M$. This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group...