Measurable solvability for measure-preserving actions

Determine whether the inclusion MEASURE(Γ)⊆MEASURE_pmp(Γ) is strict for finitely generated groups Γ.

Background

MEASURE(Γ) requires measurable solutions on every free Borel action equipped with a standard probability measure, whereas MEASURE_pmp(Γ) only requires solutions for free measure-preserving actions. The former class is automatically contained in the latter, but the paper leaves open whether non-measure-preserving actions impose strictly stronger requirements.

References

Clearly, $MEASURE() \subseteq MEASURE_{\mathsf{pmp}()}$. Can this inclusion be strict?

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Question 6.10, Section 6, subsection “Open problems”

Can this inclusion be strict?

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Question 7.10, Section 7, subsection “Open problems”