Lebesgue extension of a positive operator. (English. Russian original) Zbl 0696.47035

Akilov, G. P.; Kolesnikov, E. V.; Kusraev, A. G.

Sov. Math., Dokl. 37, No. 1, 88-91 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 3, 521-524 (1988).

Summary: All the known schemes for constructing an integral are subsumed under the following general scheme. Let X be a vector lattice and \(\Phi\) a given positive functional on it - a pre-integral. A K-space (i.e. a conditionally complete vector lattice) extending X is constructed, and the pre-integral \(\Phi\) is extended to an order-continuous functional, which is an integral. This scheme is realized here for an arbitrary positive operator \(\Phi\) with values in a K-space, i.e., for a vector- valued pre-integral. In passing, a Radon-Nikodým theorem is obtained for positive operators, along with a detailed description of the base of the Lebesgue extension.

MSC:

47B60	Linear operators on ordered spaces
47B38	Linear operators on function spaces (general)
28D05	Measure-preserving transformations
28A15	Abstract differentiation theory, differentiation of set functions
46G10	Vector-valued measures and integration
28B05	Vector-valued set functions, measures and integrals

Keywords:

schemes for constructing an integral; vector lattice; pre-integral; K- space; conditionally complete vector lattice; order-continuous functional; vector-valued pre-integral; Radon-Nikodým theorem; Lebesgue extension