Integral representation of invariant functionals
Let \(\Sigma\) be an algebra of subsets of a set \(S\) and \(B(S,\Sigma)\) the Banach space of all real-valued, totally \(\Sigma\)-measurable functions, i.e., the closure of the space of all real-valued \(\Sigma\)-simple functions with respect to the sup-norm \(\|\cdot\|_\infty\). The authors give a representation of all continuous functionals \(I: (B(S,\Sigma),\|\cdot\|_\infty)\to \mathbb{R}\) which are order invariant, i.e., which satisfy \(\Phi(I(f))= I(\Phi\circ f)\) for any order isomorphism \(\Phi:\mathbb{R}\to \mathbb{R}\) and \(f\in B(S,\Sigma)\).
scientific article; zbMATH DE number 1441399