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