On characterizations of sup-preserving functionals
Given a Banach lattice \(E\), a function \(\Lambda\colon E_+\to \mathbb R_+\) is said to be an M-functional if \(\Lambda(f\vee g)=\Lambda(f)\vee\Lambda(g)\) and \(\Lambda(\alpha f)=\alpha\Lambda(f)\) for all \(f\) and \(g\) in \(E_+\) and all \(\alpha\) in \(\mathbb R_+\). A function \(\mu : \Sigma\to\mathbb R+\) on an algebra of subsets of a set \(X\) is said to be an M-measure if \(\mu(A\cup B)= \mu(A)\vee\mu(B)\) for all \(A,B\) in \(\Sigma\). The authors show that given a set \(X\), an algebra, \(\Sigma\) of subsets \(X\) and an M-measure \(\mu\) on \(\Sigma\), the function \(\Lambda_\mu(f)=\sup_{r\in \mathbb R_+}r\mu(f^{-1}((r, \infty)))\) defines an M-functional on any vector sublattice \(E\) of the lattice of bounded real-valued functions on \(X\) for which each element of \(E_+\) is \(\Sigma\)-measurable. In addition, they show that \(\Lambda_\mu\) may also be obtained from the formula \(\Lambda_\mu(f)=\sup_{r\in \mathbb R_+}(\lim_{h\to 0^+}\mu(f^{-1}((r-h,r+h))))\) for \(f\) in \(C(X,\mathbb R)_+\). Conversely, given a topological space \(X\) and an M-functional \(\Lambda\) on \(C(X,\mathbb R)_+\), the function \(\mu(A)=\inf\{\Lambda(f):f\in C(X,\mathbb R )_+,f\geq 1 \text{ on } A\}\) defines an M-measure on the algebra of a subsets of \(X\) such that \(\Lambda(f)=\Lambda_\mu(f)\) for all \(f\) in \(C(X,\mathbb R)_+\).
scientific article; zbMATH DE number 5651863