On Schachermayer and Valdivia results in algebras of Jordan measurable sets

López-Alfonso, Salvador

doi:10.1007/s13398-015-0267-x

Cited by 11 publications

(12 citation statements)

References 8 publications

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“…The inequality (6) is a particular case of (11). Finally from (12) with i = r we get (8) because each (i np , j np ) verifies that r = i np n p j np .…”

Section: Proof Of Theoremmentioning

confidence: 95%

On Valdivia strong version of Nikodym boundedness property

Ka̧kol

Pellicer

2017

Journal of Mathematical Analysis and Applications

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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...

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“…The inequality (6) is a particular case of (11). Finally from (12) with i = r we get (8) because each (i np , j np ) verifies that r = i np n p j np .…”

Section: Proof Of Theoremmentioning

confidence: 95%

On Valdivia strong version of Nikodym boundedness property

Ka̧kol

Pellicer

2017

Journal of Mathematical Analysis and Applications

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“…, N n+1,k n+1 . At least one of this subsets, named N n+1 verifies that we get that lim s→∞ μ i ms , j ms (A) = ∞, in contradiction with (10).…”

Section: N P and For An Infinity Of Values Of Nmentioning

confidence: 86%

“…Previous related results can be found in [5,14]. An example of an algebra A such that A is a web Nikodým set for ba(A ) is given in [10].…”

Section: Nikodým Set For Ba(σ)mentioning

confidence: 99%

Vitali–Hahn–Saks property in coverings of sets algebras

Alfonso

2020

RACSAM

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A subset B of an algebra A of subsets of Ω is a Nikodým set for ba (A ) if each B-pointwise bounded subset M of ba(A ) is uniformly bounded on A and B is a strong Nikodým set for ba(A ) if each increasing covering (B m ) ∞ m=1 of B contains a B n which is a Nikodým set for ba(A ), where ba(A ) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A . The subset B has (V H S) property if B is a Nikodým set for ba(A ) and for each sequence (μ n ) ∞ m=1 and each μ, both in ba(A ) and such that lim n→∞ μ n (B) = μ(B), for each B ∈ B, we have that the sequence (μ n ) ∞ m=1converges weakly to μ. We prove that if (B m ) ∞ m=1 is an increasing covering of and algebra A that has (V H S) property and there exist a B n which is a Nikodým set for ba(A ) then there exists B q , with q ≥ p, such that B q has (V H S) property. In particular, if (B m ) ∞ m=1 is an increasing covering of a σ -algebra there exists B q that has (V H S) property. Valdivia proved that every σ -algebra has strong Nikodým property and in 2013 asked if Nikodým property in an algebra implies strong Nikodým property. We present three open questions related with this aforementioned Valdivia question and a proof of his strong Nikodým Theorem for σ -algebras that it is independent of the Barrelled spaces theory and it is developed with basic results of Measure theory and Banach spaces. Keywords Bounded set • Algebra and σ -algebra of subsets • Bounded finitely additive scalar measure • Nikodym and strong Nikodym property • Vitali-Hahn-Saks and strong Vitali-Hahn-Saks property

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“…Koszmider and Shelah have shown in [13] that if an infinite algebra A has the so-called Weak Subsequential Separation Property then the cardinal of A is greater than or equal to the continuum c. Since all algebras considered here have the Weak Subsequential Separation Property, it arises the natural question whether there exist algebras with the Nikodým property with cardinality less than c. This question has been solved positively by Sobota in [25]. On the other hand, in [14,Theorem 1] it was proved that the algebra J (K) of Jordan measurable subsets of the compact interval [27,Theorem 4]. Note that |J (K)| = 2 c , where |A| denotes the cardinality of the set A. Valdivia asked in [27,Problem 1] whether the equivalence (N) ⇔ (sN) holds for an algebra A of sets which is not a σ-algebra.…”

Section: Preliminariesmentioning

confidence: 97%

On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

Ferrando

López-Alfonso

Pellicer

2019

Filomat

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If R is a ring of subsets of a set Ω and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily ∆ of R is called a Nikodým set for ba (R) if each set {µ α : α ∈ Λ} in ba (R) which is pointwise bounded on ∆ is norm-bounded in ba (R). If the whole ring R is a Nikodým set, R is said to have property (N), which means that R satisfies the Nikodým-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck's property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikodým sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N) ⇔ (wN) holds.

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On Schachermayer and Valdivia results in algebras of Jordan measurable sets

Cited by 11 publications

References 8 publications

On Valdivia strong version of Nikodym boundedness property

On Valdivia strong version of Nikodym boundedness property

Vitali–Hahn–Saks property in coverings of sets algebras

On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

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