General existence conditions for Lebesgue decompositions in CMon-enriched/semiadditive CD categories

Establish general conditions that guarantee the existence of Lebesgue decompositions for morphisms in CMon-enriched (semiadditive CD) categories, and develop an appropriate Radon–Nikodym theorem; in particular, test the conjecture that completeness of the enrichment-induced preorder ≤ is sufficient to obtain such a theorem and decomposition existence.

Background

Earlier in the paper, the authors define an abstract Lebesgue decomposition P = P_ac + P_s in CMon-enriched categories using the absolute continuity preorder (≪) and singularity (⊥) via meets, and show that this abstraction recovers the classical measure-theoretic notion when instantiated in categories of kernels.

While they establish structural properties of decompositions and use them to generalize results on Metropolis–Hastings, they do not provide general existence theorems for such decompositions in the enriched categorical setting. In the conclusion, they identify establishing existence as a key open direction and conjecture that an appropriate Radon–Nikodym theorem may be attainable under completeness of the ≤ preorder.

References

An interesting direction for future work is to give general conditions under which Lebesgue decompositions exist in this setting, rather than only isolating the properties that such decompositions ought to satisfy. This appears to require a suitable version of the Radon--Nikodym theorem, which we conjecture may be attainable by assuming that the preorder ≤ induced by the enrichment is complete.

— A categorical account of the Metropolis-Hastings algorithm  (2601.22911 - Cornish et al., 30 Jan 2026) in Section 6 (Conclusion and future work)