Absolute continuity characterization sets

Abstract:

Suppose $U$ is a set, ${\mathbf {F}}$ is a field of subsets of $U,{\mathfrak {p}_{AB}}$ is the set of all real-valued, bounded finitely additive functions on ${\mathbf {F}}$, and for each $\rho$ in ${\mathfrak {p}_{AB}},{\mathcal {A}_\rho }$ is the set of all elements of ${\mathfrak {p}_{AB}}$ absolutely continuous with respect to $\rho ,\mathfrak {p}_A^ +$ is the set of all nonnegative-valued elements of ${\mathfrak {p}_{AB}}$, and ${\mathfrak {p}_B}$ is the set of all functions from ${\mathbf {F}}$ into $\exp ({\mathbf {R}})$ with bounded range union. An extension of a previous absolute continuity characterization theorem of the author (Proc. Amer. Math. Soc. 18 (1967), 94-99) is given in the form of a characterization of those subsets $S$ of ${\mathfrak {p}_{AB}}$ having the property that if each of $\xi$ and $\mu$ is in $\mathfrak {p}_A^ +$, then $\xi$ is in ${\mathcal {A}_\mu }$ iff it is true that if $\alpha$ is in ${\mathfrak {p}_B},{\smallint _U}\alpha (I)\mu (I)$ and ${\smallint _U}\alpha (I)\xi (I)$ exist and the function $\smallint \alpha \mu$ is in $S$, then $\smallint \alpha \xi$ is in $S$.