Journal of Mathematical Analysis and Applications 324 (2006) 862-881. doi:10.1016/j.jmaa.2005.12.059Received by publisher: 2005-04-22Harvest Date: 2016-01-04 12:20:16DOI: 10.1016/j.jmaa.2005.12.059Page Range: 862-88
Following Schachermayer, a subset B of an algebra A of subsets of Ω is said to have the N -property if a B-pointwise bounded subset M of ba(A) is uniformly bounded on A, where ba(A) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A. Moreover B is said to have the strong N -property if for each increasing countable covering (B m ) m of B there exists B n which has the N -property. The classical Nikodym-Grothendieck's theorem says that each σ-algebra S of subsets of Ω has the N -property. The Valdivia's theorem stating that each σ-algebra S has the strong N -property motivated the main measure-theoretic result of this paper: We show that if (B m1 ) m1 is an increasing countable covering of a σ-algebra S and if (B m1,m2,...,mp,mp+1 ) mp+1 is an increasing countable covering of B m1,m2,...,mp , for each p, m i ∈ N, 1 i p, then there exists a sequence (n i ) i such that each B n1,n2,...,nr , r ∈ N, has the strong Nproperty. In particular, for each increasing countable covering (B m ) m of a σ-algebra S there exists B n which has the strong N -property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided. *Manuscriptis the Minkowski functional of absco({χ C : C ∈ A}) ([14, Propositions 1 and 2]) and its dual norm is the A-supremum norm, i.e., µ := sup{|µ(C)| : C ∈ A}, µ ∈ ba(A). The closure of a set is marked by an overline, hence if P ⊂ L(A) then span(P ) is the closure in L(A) of the linear hull of P . N is the set {1, 2, . . .} of positive integers.Recall the classical Nikodym-Dieudonné-Grothendieck theorem (see [1, page 80, named as Nikodym-Grothendieck boundedness theorem]): If S is a σ-algebra of subsets of a set Ω and M is a S-pointwise bounded subset of ba(S) then M is a bounded subset of ba(S) (i.e., sup{|µ(C)| : µ ∈ M, C ∈ S} < ∞, or, equivalently, sup{|µ| (Ω) : µ ∈ M } < ∞). This theorem was firstly obtained by Nikodym in [11] for a subset M of countably additive complex measures defined on S and later on by Dieudonné for a subset M of ba(2 Ω ), where 2 Ω is the σ-algebra of all subsets of Ω, see [3].It is said that a subset B of an algebra A of subsets of a set Ω has the Nikodym property, N -property in brief, if the Nikodym-Dieudonné-Grothendieck theorem holds for B, i.e., if each B-pointwise bounded subset M of ba(A) is bounded in ba(A) (see [12, Definition 2.4] or [15, Definition 1]). Let us note that in this definition we may suppose that M is τ s (A)-closed and absolutely convex. If B has N -property then the polar setIt is well known that the algebra of finite and co-finite subsets of N fails N -property [2, Example 5 in page 18] and that Schachermayer proved that the algebra J (I) of Jordan measurable subsets of I := [0, 1] has N -property (see [12, Corollary 3.5] and a generalization in [4, Corollary]). A recent improvement of this result for the algebra J (K) of Jordan measurable subsets of a compact k-dimensional interval K := Π{[a i , b i ] : 1 i k} in R k has been provi...
Elsevier Gabriyelyan, S.; Kakol, JM.; Kubzdela, A.; López Pellicer, M. (2015). On topological properties of Fréchet locally convex spaces. Topology and its Applications. 192 AbstractWe describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ(E, E ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) 1 and satisfying the Heinrich density condition we prove that (E, σ(E, E )) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain 1 , then (E, σ(E, E )) is an ℵ 0 -space if and only if E is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E )) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF )-space whose strong dual E is separable, then (E, σ(E, E )) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for ǎ Cech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C c (X) is cosmic in the weak topology, (c) the weak * -dual of C c (X) is an ℵ 0 -space.
Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space C p (X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff C p (X) is Baire (i.e. of the second Baire category) and is covered by a family {K α : α ∈ N N } of compact sets such that K α ⊂ K β if α β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family {K α : α ∈ N N } of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.
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