The standard change of variable formula, for measurable from a measure space to a measurable space on which a measurable is defined (see e.g.[H]), , is developed for a semigroup valued refinement integral. (The integral on the right can be rewritten in “Jacobian” form, , when is differentiable with respect to a measure on ; the attempt in [DS] Lemma III.10.8 to develop the result by starting from is incorrect). This yields a result for the order-convergent integral of real-valued integrands against positive finitely additive measures in an Archimedian ordered vector space as well as a (correct) result for the [DS] Banach space valued integral of a vector valued integrand against a real-valued finitely additive measure.
"Change of variable in the semigroup valued refinement integral." Real Anal. Exchange 15 (1) 106 - 110, 1989/1990. https://doi.org/10.2307/44151995