Reversible subsets of certain linear transformations (Q1821217)

Suppose U is a set, \({\mathbb{F}}\) is a field of subsets of U, \(p_{AB}\) is the set of all real-valued bounded finitely additive functions defined on \({\mathbb{F}}\), and for each \(\rho\) in \(p_{AB}\), \(A(\rho)=\{\xi:\xi\) in \(p_{AB},\xi\) absolutely continuous with respect to \(\rho\}\). Suppose M is a linear subspace of \(p_{AB}\) such that \(M=\cup_{\eta in M}A(\eta).\) A generalization of a previously discussed collection of linear transformations [see the author, J. Lond. Math. Soc. 44, 385-396 (1969; Zbl 0169.068)] is treated by letting \(C_ M\) denote the set to which T belongs iff T is a linear transformation from M into \(p_{AB}\) such that for some K in \({\mathbb{R}}\) and all \(\xi\) in M and V in \({\mathbb{F}}\), \(\int_{V}| T(\xi)(I)| \leq K\int_{V}| \xi (I)|.\) Certain theorems of the aforementioned reference are generalized, as well as one from the author's paper [Trans. Am. Math. Soc. 199, 131-140 (1974; Zbl 0289.28006)]. The principal result of the present paper is the following generalization of a reversibility characterization in the first mentioned reference: Theorem: If T is in \(C_ M\), then \(\{(\eta,T(\eta)):\eta \quad in\quad M\cap A(T(\eta))\}\) is the only reversible subset \(T^ 0\) of T such that: i) the domain \(M^ 0\) of \(T^ 0\) is a linear subspace of M and \(M^ 0=\bigcup_{\eta\text{ in }M^ 0}A(\eta)\), and ii) the range of \(T^ 0\) is the range of T.