On the question of universal integrability of bounded functions

A. G. Chentsov

Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 75-94

Résumé

Various procedures are considered for integration of bounded real-valued functions with respect to finitely additive measures on a semi-algebra of sets. A criterion is established for the indefinite Darboux integrals of a bounded function to coincide on the family of all positive finitely additive measures: a bounded function is universally integrable if and only if it belongs to the closure, in the metric of uniform convergence, of the linear span of the family of all characteristic functions of sets in the semi-algebra. A representation of the indefinite Darboux integral is obtained for such bounded functions. For arbitrary bounded functions a construction is proposed for a multivalued indefinite integral with respect to a positive finitely additive measure, and some of the properties of the construction are established. In particular, the multivalued integral of an arbitrary bounded function with respect to a positive countably additive measure consists only of countably additive measures of bounded variation, while the multivalued integral with respect to a purely finitely additive positive measure consists only of purely finitely additive measures. The dependence of the multivalued integral on the bounded function is continuous in the sense of the natural metric for the space of nonempty order intervals in the family of finitely additive measures of bounded variation. Bibliography: 7 titles.