Deterministic Transport Under Mutual Multiphase Domination

Definitions and mathematical statement

Write the supplied vector-valued measure spaces as $(X,\mathcal A,\sigma)$ and $(Y,\mathcal B,\eta)$, where $\mathcal A$ and $\mathcal B$ denote their measurable structures and $\sigma=(\sigma_i)_{i\in I}$ and $\eta=(\eta_i)_{i\in I}$ denote their component measures. The common component index set $I$, the permitted types of component measures, and the precise meaning of atomlessness are not specified in the supplied record. Retain the source's assumption that both vector-valued measure spaces are atomless, subject to these unresolved qualifications.

For a measurable map $T:X\to Y$, define its componentwise pushforward by \[ (T_\#\sigma_i)(B)=\sigma_i(T^{-1}(B))\qquad(B\in\mathcal B). \] For a measurable map $S:Y\to X$, similarly define \[ (S_\#\eta_i)(A)=\eta_i(S^{-1}(A))\qquad(A\in\mathcal A). \] Write $\preceq_{\mathrm{mp}}$ solely as a placeholder for the source's multiphase-transport order; the supplied record gives no definition of this relation.

The proposed question is whether, for every pair of atomless vector-valued measure spaces in the intended class, the mutual-domination assumptions \[ (X,\mathcal A,\sigma)\preceq_{\mathrm{mp}}(Y,\mathcal B,\eta) \quad\text{and}\quad (Y,\mathcal B,\eta)\preceq_{\mathrm{mp}}(X,\mathcal A,\sigma) \] imply the existence of measurable maps $T:X\to Y$ and $S:Y\to X$ such that \[ \forall i\in I,\qquad T_\#\sigma_i=\eta_i\quad\text{and}\quad S_\#\eta_i=\sigma_i. \] Equivalently, the required equalities are \[ \begin{aligned} \sigma_i(T^{-1}(B))&=\eta_i(B)&& (i\in I,\ B\in\mathcal B),\\ \eta_i(S^{-1}(A))&=\sigma_i(A)&& (i\in I,\ A\in\mathcal A). \end{aligned} \] The same map $T$ must transport every component simultaneously, and likewise for $S$. The displayed source statement does not require $S\circ T$ or $T\circ S$ to be an identity, even almost everywhere. The background's phrase "deterministic inverse transports" leaves a possible stronger intended requirement unresolved. Without definitions of the order and the admissible measure spaces, this remains a qualified statement template requiring mathematical source review.

Short English statement

Does mutual domination in the intended multiphase-transport order between two atomless vector-valued measure spaces imply the existence of deterministic measurable transports in both directions, each preserving all component measures simultaneously? The supplied statement asks for maps in both directions; whether they must also be inverses is unclear from the background. The order, measure-space hypotheses, and atomlessness convention require clarification.

Sources

• Supplied local problem-registry record: data/unsolvedmath/problems.json, permanent catalogue entry, source problem code catalogue entry, title Deterministic Transport Under Mutual Multiphase Domination, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2017; no individual proposer is identified.
• Supplied report citation: Applications of Optimal Transportation in the Natural Sciences (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/7, pp. 339--416. https://doi.org/10.4171/owr/2017/7. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The citation's year 2018 and the report designation 7/2017 are retained as supplied; their relationship is not established here.
• Supplied literature source: Oberwolfach Report 7/2017, https://ems.press/content/serial-article-files/46673.
• Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified assessment. The supplied status is open, with the assessment that no general theorem had been found showing that mutual multiphase domination yields deterministic inverse transports preserving every component, and that the question remained open beyond special cases. The record explicitly calls this dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified; its accompanying explanation describes repairs to notation, typography, and sentence structure while preserving mathematical content. This supplied editorial label does not establish independent mathematical verification of this draft or of the literature assessment.