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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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##### References:
  P. De Pagter, ?The components of a positive operator,? Indag. Math.,86, 229-240 (1983). · Zbl 0521.47018  W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, Vol. 1, Amsterdam (1971). · Zbl 0231.46014  I. I. Shamaev, ?On countable extensions of measures with values in a vector lattice,? Sib. Mat. Zh.,22, No. 3, 197-203 (1981).  L. V. Kantorovich, B. Z. Vulikh, and A. G. Pinsker, Functional Analysis in Semiordered Spaces [in Russian], Gostekhizdat, Moscow (1950). · Zbl 0037.07201  K. Matthes, ?Über die Ausdehnung positiver linearer Abbildungen,? Math. Nachr.,23, 223-257 (1961). · Zbl 0104.08801  J. D. M. Wright, ?The measure extension problem for vector lattices,? Ann. Inst. Fourier,21, No. 4, 65-85 (1971). · Zbl 0215.48101  D. H. Fremlin, ?A direct proof of the Matthas-Wright integral extension theorem,? J. London Math. Soc.,11, No. 2, 276-284 (1975). · Zbl 0313.06016 · doi:10.1112/jlms/s2-11.3.276  W. A. J. Luxemburg and A. R. Schep, ?An extension theorem for Riesz homomorphisms,? Indag. Math.,41, 145-154 (1979). · Zbl 0425.46006  Z. Lipecki, D. Plachky, and W. Thomsen, ?Extensions of positive operators and extreme points. I,? Colloq. Math.,42, 279-284 (1979). · Zbl 0432.47018  Z. Lipecki and W. Thomsen, ?Extensions of positive operators and extreme points. IV,? Colloq. Math..46, 269-273 (1982). · Zbl 0432.47021  T. K. Y. C. Dodds, ?Order topology and the Egoroff property in Riesz spaces,? Not. Am. Math. Soc.,16, No. 4, 644-645 (1969).  T. K. Y. C. Dodds, ?Egoroff properties and the order topology in Riesz spaces,? Trans. Am. Math. Soc.,187, No. 1, 365-375 (1974). · Zbl 0254.46008 · doi:10.1090/S0002-9947-1974-0336282-3  I. P. Kostenko, ?Almost uniform convergence in K-spaces,? Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 49-60 (1968). · Zbl 0182.16301  D. A. Vladimirov, ?On completeness of a semiordered space,? Usp. Mat. Nauk,15, No. 2, 165-172 (1960). · Zbl 0097.09003  I. I. Shamaev, ?Some order-algebraic properties of vector lattices and their connections,? Preprint, Math. Inst. Siberian Div., USSR Acad. of Sciences, Novosibirsk (1980).
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