Change of variable in the semigroup valued refinement integral

Abstract

The standard change of variable formula, for $T$ measurable from a measure space $(S,S,\mathrm{\lambda })$ to a measurable space $(\overline{S},\overline{S})$ on which a measurable $\overline{t}$ is defined (see e.g.[H]), ${\displaystyle {\int }_S\overline{t}Td\mathrm{\lambda }}={\displaystyle {\int }_{\overline{S}}\overline{t}d(\mathrm{\lambda }{T}^{-1}})$, is developed for a semigroup valued refinement integral. (The integral on the right can be rewritten in “Jacobian” form, ${\displaystyle {\int }_{\overline{S}}\overline{t}\frac{d\mathrm{\lambda }{T}^{-1}}{d\overline{\mathrm{\lambda }}}}d\overline{\lambda }$, when $\lambda T^{-1}$ is differentiable with respect to a measure $\overline{\mathrm{\lambda }}$ on $\overline{S}$; the attempt in [DS] Lemma III.10.8 to develop the result by starting from $\overline{\mathrm{\lambda }}$ 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.