We initiate the effective metric structure theory of Keisler randomizations. We show that a classical countable structure $\mathcal{ M}$ has a decidable presentation if and only if its Borel randomization $\mathcal{ M}^{ [0,1)}$ has a computable presentation for which the constant functions are uniformly computable points. We determine a sufficient condition for which the uniform computability of the constant functions can be dropped. We show that when $\mathcal{ M}$ is effectively $\omega$-categorical, then $\mathcal{ M}^{ [0,1)}$ is computably categorical, that is, has a unique computable presentation up to computable isomorphism. A special case of this result is that the unique separable atomless probability algebra is computably categorical. Finally, we show that all randomizations admit effective quantifier elimination.
Categories: math.LO