Dominated Convergence, Fubini–Tonelli and Iterated Trace Interchange for Measurable Endofunctor Families. Trace Reciprocity III: Product Measures and Limit Interchange

This paper develops the product-measure and convergence layer of the Trace Reciprocity Framework. Starting from the Banach-space trace-realization architecture established in Trace Reciprocity II, it studies two-parameter measurable families, iterated Bochner integration, order independence, and convergence under limits. Under product trace-norm integrability, Banach-valued Fubini–Tonelli theory identifies the product integral with both orders of iterated integration, while the bounded graded trace functional produces the corresponding order-independent Trace Reciprocity identity. The paper also establishes dominated convergence at the realization level, countable-series interchange, weighted product decompositions, and finite-depth hierarchical reciprocity. Explicit counterexamples demonstrate why scalar trace integrability alone does not guarantee existence of a Bochner-integrated realization and why absolute control is necessary for order-independent reconstruction. Trace Reciprocity Framework — Paper III.