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Countable extension of measures and \(\sigma\)-integrals with values in vector lattices. (English. Russian original) Zbl 0614.28004
Math. Notes 39, 414-418 (1986); translation from Mat. Zametki 39, No. 5, 757-765 (1986).
A \(K_{\sigma}\)-space (i.e., a Dedekind \(\sigma\)-complete linear lattice) Y has the property of weak \(\sigma\)-distributivity of countable type if every order bounded double sequence \(\{y_{nm}\}\) in Y which, for each fixed n, is increasing in m, admits an increasing map \(\theta:N\to N^ N\) with \[ \sup_{n}\inf_{m}y_{nm}=\inf_{k}\sup_{n}y_{n\theta_ k}(n). \] In an earlier paper [Sib. Mat. Zh. 22, 197-203 (1981; Zbl 0482.28016)] the author introduced this property and proved it to be equivalent to some properties of Y which are expressed in measure-theoretic and topological terms. Here he supplements those results by showing that the property in question is equivalent to the Egorov property as well as to the existence of a special extension for every Y-valued \(\sigma\)-integral defined on a majorizing linear sublattice of another \(K_{\sigma}\)- space.
{Reviewer’s remarks: (1) Some arguments are missing and some others are sketchy. In particular, more details in the proof of the equivalence (v)\(\Leftrightarrow (vi)\) would have been in order. (2) There is a repairable mistake in the proof the implication (ii)\(\Rightarrow (iii)\). Namely, \(G_{x_ n}\uparrow G_ x\) does not hold, even for constant functions. (3) For related results see M. Vonkomerová [Math. Slovaca 31, 251-262 (1981; Zbl 0457.46004)].}
Reviewer: Z.Lipecki
MSC:
28B15 Set functions, measures and integrals with values in ordered spaces
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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