INTERVALS OF FINITELY ADDITIVE SET FUNCTIONS

Abstract

Suppose that $\text{U}$ is a set, $\text{F}$ is a field of subsets of $\text{U}$, $\text{A}(ℝ)(\text{F})$ is the set of all real – valued finitely additive functions defined on $\text{F}$, $\text{A}(ℝ)(\text{F}{)}^{+}$ is the set of all nonnegative – valued elements of $\text{A}(ℝ)(\text{F})$, each of ${\mathrm{\xi }}_1$ and ${\mathrm{\xi }}_2$ is in $\text{A}(ℝ)(\text{F})$, ${\mathrm{\xi }}_2-{\mathrm{\xi }}_1$ is in $\text{A}(ℝ)(\text{F}{)}^{+}$, $\alpha$ is a function with domain $\text{F}$ and range a collection of subsets of $ℝ$ with bounded union, and for $\text{i}=1,2$, the integral ${\int }_{\text{U}}\mathrm{\alpha }{\xi }_{\text{i}}$, as a refinement – wise limit of sums, exists. Let $\text{T}$ denote the transformation with domain $\{\rho :\rho \text{ in A}(ℝ)(\text{F}),\text{ each of }{\xi }_2-\rho and\rho -{\xi }_1\text{ in A}(ℝ){(\text{F})}^{+}\}$ and range $\subseteq \text{A}(ℝ)(\text{F})$ given by $\text{T}(\rho )(V)={\int }_{\text{V}}\alpha \rho$. Continuity, closure, maximum value, minimum value and convergence properties of T are studied.