On regular extensions of contents and measures
Let \({\mathcal K}\) be a lattice of subsets of X. Let N(\({\mathcal K})\) be the family of all [0,\(\infty]\)-valued set functions defined on \({\mathcal K}\) and vanishing at \(\emptyset\). If \(\lambda \in N({\mathcal K})\) we say \(\lambda\) to be modular if \[ \lambda (k_ 1)+\lambda (k_ 2) = \lambda (k_ 1\cap k_ 2)+\lambda (k_ 1\cup k_ 2) \] and \(\lambda\) is said to be supermodular if \[ \lambda (k_ 1)+\lambda (k_ 2)\leq \lambda (k_ 1\cap k_ 2)+\lambda (k_ 1\cup k_ 2). \] It is said to be tight if \[ \lambda (k_ 2) = \lambda (k_ 1)+Sup\{\lambda (k)| \quad k\in {\mathcal K},\quad k\subset k_ 2-k_ 1\} \] for \(k_ 1,k_ 2\in {\mathcal K}\) with \(k_ 1\subset k_ 2\). Based on the concept of tight set function and the fact that every supermodular set function defined on a lattice of sets admits a tight majorant, the author generalizes extension theorems due to others including that of his own [Trans. Am. Math. Soc. 283, 353-368 (1984; Zbl 0508.28001)].
scientific article; zbMATH DE number 4049926
scientific article