min and max functions, Bochner-Radon-Nikodým approximations, and a representation characterization
Basic properties of min\(\{\) x,y\(\}\), max\(\{\) x,y\(\}\), the sum supremum and sum infimum functionals and differential equivalence are used to obtain the following for the refinement integral: (1) a form of the Bochner-Radon-Nikodým theorem (a finitely additive \(\mu\)-continuous measure can be approximated arbitrarily close in variation norm by an indefinite \(\mu\)-integral) for set-valued set-functions which does not require integrability, and (2) a characterization of real-valued functions h with domain [a,b] which admit a representation of the form \[ h(q)-h(p)=\int \max \{\min \{\alpha (I)-p,q-p\},0\}\mu (I),\quad a\leq p\leq q\leq b, \] for some nonnegative finitely additive function on a field of sets and some set-valued function \(\alpha\) on the same field. The characterization in (2) is a bounded concavity condition: \[ M\geq (h(t)-h(s))/(t-s)\geq (h(u)-h(t))/(u-t)\geq 0 \] for all \(a\leq s<t<u\leq b\).
scientific article; zbMATH DE number 4146317
scientific article