Recovering a presentation of M from a presentation of its Borel randomization

Determine whether one can, in general, extract a presentation (decidable or otherwise) of a countable classical structure M from an arbitrary presentation of its Borel randomization M^{[0,1)}; explicitly develop a method to reconstruct a presentation of M given only a presentation of M^{[0,1)}.

Background

The paper constructs an induced computable presentation of the Borel randomization M^{[0,1)} starting from a decidable presentation of M and proves several equivalences under additional assumptions. However, going in the reverse direction—from a presentation of the randomization back to a presentation of M—poses methodological difficulties.

This problem asks for a general procedure to recover a presentation of M from any given presentation of M^{[0,1)} without additional assumptions such as awareness or uniform access to constant functions.