Characterize the domain of the extension for metrizable target groups

Determine whether the equality D_ν = \mathcal{S}_A[μ] holds whenever A is metrizable, where A is a sequentially complete Hausdorff topological group, μ is a weakly exhaustive A-valued σ-content, ν is the unique A-valued measure extending μ, and \mathcal{S}_A[μ] is the σ-ring defined in equation (\ref{eq:Smu}).

Background

Theorem \ref{t_ext} constructs a unique A-valued measure ν extending a weakly exhaustive A-valued σ-content μ and proves the inclusions D_μ ⊂ \mathcal{S}A[μ] ⊂ Dν. The paper gives an example in Appendix \ref{s_example2} where the second inclusion is proper, so equality does not hold without additional assumptions.

The authors identify metrizability of the target group A as a possible sufficient condition for equality, but do not establish it. Resolving this question would provide an exact description of the domain of the extension measure in an important class of topological groups.

References

In general, it may happen that $D_\nu \neq \mathcal{S}{#1 A}[\mu]$ under the conditions of Theorem~\ref{t_ext} (see the example in Appendix~\ref{s_example2}). We expect that the equality $D\nu = \mathcal{S}_{#1 A}[\mu]$ holds for metrizable~$#1 A$.

— Measure theory without infinities  (2609.03875 - Smirnov et al., 3 Sep 2026) in Section 10, immediately after Theorem 10.1 (Theorem \ref{t_ext}); Appendix \ref{s_example2} provides an example showing that the inclusion can be proper in general.