Explicit description of a right adjoint (Borel locale) to the inclusion of measurable locales into locales

Determine an explicit construction and description of the right adjoint (if it exists) to the inclusion functor i: MblLoc → Loc from measurable locales to locales. Concretely, construct the hypothesized functor (-)_Bor: Loc → MblLoc that assigns to each locale L its "Borel locale" and provide its defining properties and universal characterization, thereby giving an explicit account of the right adjoint to i.

Background

Measurable locales (MblLoc) are Boolean locales equipped with sufficiently many measures, and they are dual (via a Gelfand-type duality) to commutative von Neumann algebras. There is a natural inclusion functor from measurable locales into the broader category of locales (Loc).

The author suggests it is plausible that this inclusion might admit a right adjoint, which would assign to any locale a canonical measurable locale analogously thought of as a "Borel locale." Such a functor would systematically extract measurable structure from arbitrary locales and would refine the categorical foundations of measure theory in the localic setting.

The remark contrasts this potential right adjoint with the situation for Boolean locales, where the inclusion BoolLoc → Loc is known not to admit such a right adjoint in general. Despite plausibility of existence for MblLoc, an explicit description or construction of the putative right adjoint functor remains unspecified.