Measurable Families and Trace Interchange for Trace-Class Endofunctors: Foundations of Trace Reciprocity II

Measurable Families and Trace Interchange for Trace-Class Endofunctors develops the functional-analytic foundations of the Trace Reciprocity Framework. Building on the introductory Trace Reciprocity result, the paper replaces schematic measurable decomposition with a precise framework based on trace realizations into a graded ℓ¹-sum of trace-class operator ideals. Measurability is formulated through strong Bochner measurability, while integration is performed in trace norm. The resulting graded trace is a bounded linear functional, allowing the principal Trace Reciprocity identity to follow from continuity of trace under Bochner integration. The paper establishes approximation by simple realizations, dominated convergence, discrete and weighted forms, and local-to-global partition reconstruction. Particular attention is given to the distinction between Bochner integration in a fixed Banach space and direct-integral decompositions of Hilbert spaces. The paper identifies the appropriate von Neumann-algebraic setting for the latter and separates realized trace, ordinary operator trace, and intrinsic categorical trace. Trace Reciprocity Framework — Paper II.