Geometric properties of the range of two-dimensional quasi-measures with respect to the Radon-Nikodým property
Let \(\mu\) and \(\nu\) be nonnegative, finite quasi-measures (finitely additive measures) on a \(\sigma\)-field \({\mathcal A}\) of subsets of a set \(\Omega\). In addition, assume that \(\mu\) and \(\nu\) are continuous in the sense that for each \(\varepsilon>0\) there is a finite partition \(\{A_ 1,A_ 2,\dots,A_ n\}\) of \(\Omega\) such that \(A_ i\in{\mathcal A}\) and \(\mu(A_ i)<\varepsilon\) for each \(i\), and similarly for \(\nu\). This paper investigates existence of a Radon-Nikodým derivative of \(\nu\) with respect to \(\mu\), that is, a nonnegative integrable function \(f\) on \(\Omega\) for which \(\nu(E)=\int_ Ef d\mu\) for all \(E\in{\mathcal A}\), from the perspective of the range \(R=\{(\mu(E),\nu(E)):E\in{\mathcal A}\}\). When such a function \(f\) exists, \((\mu,\nu)\) is called an \textit{integral} quasi-measure. In this setting, there are nondecreasing functions \(G\) and \(g\), defined on the range of \(\mu\), which form the upper and lower boundaries of \(R\). The upper function \(G\) is concave while the lower function \(g\) is convex. Assuming \(\nu\ll\mu\), it is shown that ``\(R\) is closed'' is a sufficient but not necessary condition for \((\mu,\nu)\) to be integral. More generally, a condition on which points of the boundary lie in \(R\) is shown to be necessary and sufficient for \((\mu,\nu)\) to be integral. Finally, it is shown that any concave, nondecreasing function \(G:[0,a]\to[0,\infty)\) with \(G(0)=0\) determines the upper boundary of the range of an integral quasi-measure.
scientific article; zbMATH DE number 62678
scientific article