Awareness of computable presentations of Borel randomizations

Determine whether every computable presentation of the Borel randomization M^{[0,1)} is aware; that is, ascertain whether the subset of constant functions (identified with M) is always a computably enumerable closed set with respect to any computable presentation of M^{[0,1)}.

Background

A presentation of M^{[0,1)} is called aware if the embedded copy of M (as constant random variables) forms a computably enumerable closed subset relative to the presentation. The paper shows awareness for induced presentations coming from weakly computable presentations of M and gives characterizations connecting awareness with effective access to the constants.

The general status of awareness for arbitrary computable presentations of M^{[0,1)} is unclear; the authors suspect a negative answer but lack a counterexample.