Develop the Jordan decomposition and order theory of infinite signed measures

Establish a Jordan decomposition for the ordered vector space M(\mathcal{Q},R) of infinite signed measures, characterize its conditional-completeness order structure, and determine how the associated-measure operation interacts with the completion procedures described in SmirnovCompl, including whether these operations essentially commute.

Background

The paper constructs the ordered vector space M(\mathcal{Q},R) of infinite signed measures and develops its algebraic and order-theoretic foundations. It notes that this space is in fact a conditionally complete lattice, but does not prove a Jordan decomposition or fully develop the resulting order structure.

The associated-measure operation is also presented as a foundation for integration theory and Radon–Nikodym results. The authors identify the interaction between this operation and completion procedures as another unresolved topic, with the expectation that the operations essentially commute.

References

These topics will be considered in a subsequent paper. Other problems that we plan to address include the Jordan decomposition in the space $#1 M(\mathcal{Q},)$, the order structure of this space (which is actually a conditionally complete lattice), and the relationship between the associated-measure operation and various completion procedures described in (they are expected to essentially commute).

— Measure theory without infinities  (2609.03875 - Smirnov et al., 3 Sep 2026) in Section 1, Introduction, paragraph beginning “The associated-measure operation also provides the basic extension step”