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

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2409-2416 ◽  
Author(s):  
J.C. Ferrando ◽  
S. López-Alfonso ◽  
M. López-Pellicer
Keyword(s):  
Banach Space ◽  
Partial Solution ◽  
Supremum Norm ◽  
Boundedness Theorem ◽  
Finitely Additive Measures ◽  
Additive Measures

If R is a ring of subsets of a set ? 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 Nikod?m set for ba (R) if each set {?? : ???} in ba (R) which is pointwise bounded on ? is norm-bounded in ba (R). If the whole ring R is a Nikod?m set, R is said to have property (N), which means that R satisfies the Nikod?m-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 Nikod?m 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.

scholarly journals On Four Classical Measure Theorems

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 526
Author(s):  
Salvador López-Alfonso ◽  
Manuel López-Pellicer ◽  
Santiago Moll-López
Keyword(s):  
Banach Space ◽  
Measure Theory ◽  
Bounded Sequence ◽  
Survey Paper ◽  
Finitely Additive Measures ◽  
Barrelled Spaces ◽  
Type Property ◽  
Variation Norm ◽  
Additive Measures ◽  
Locally Convex

A subset B of an algebra A of subsets of a set Ω has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B-pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering {Bn:n∈N} of B contains a set Bp with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn1n2⋯nm:ni∈N,1≤i≤m,m∈N} of B contains a strand {Bp1p2⋯pm:m∈N} formed by elements Bp1p2⋯pm with property (N) [(G) or (VHS)] for every m∈N. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every σ-algebra has properties (N), (sN), (G) and (VHS). Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every σ-algebra has property (wN) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property (wN) of a σ-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (wWHS) if and only if B has property (wN) and A has property (G).

scholarly journals On regular and sigma-smooth two valued measures and lattice generated topologies

1993 ◽  
Vol 16 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Robert W. Shutz
Keyword(s):  
Finitely Additive Measures ◽  
Additive Measures

LetXbe an abstract set andLa lattice of subsets ofX.I(L)denotes the non-trivial zero one valued finitely additive measures onA(L), the algebra generated byL, andIR(L)those elements ofI(L)that areL-regular. It is known thatI(L)=IR(L)if and only ifLis an algebra. We first give several new proofs of this fact and a number of characterizations of this in topologicial terms.Next we consider,I(σ*,L)the elements ofI(L)that areσ-smooth onL, andIR(σ,L)those elements ofI(σ*,L)that areL-regular. We then obtain necessary and sufficent conditions forI(σ*,L)=IR(σ,L), and in particuliar ,we obtain conditions in terms of topologicial demands on associated Wallman spaces of the lattice.

Commutative L-algebras and measure theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Measure Theory ◽  
Measurable Space ◽  
Universal Covering ◽  
Continuous Functions ◽  
Stone Space ◽  
Integration Theory ◽  
General Measure ◽  
Lattice Order ◽  
Finitely Additive Measures ◽  
Additive Measures

Abstract Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

Parallelisability in Banach spaces: a parallelisable dynamical system with uniformly bounded trajectories

1988 ◽  
Vol 108 (3-4) ◽  
pp. 371-378
Author(s):  
B. M. Garay
Keyword(s):  
Dynamical System ◽  
Banach Space ◽  
Banach Spaces ◽  
Supremum Norm ◽  
Uniformly Bounded ◽  
Bounded Trajectories

SynopsisIn the Banach space of real sequences which converge to zero with the supremum norm, we construct a parallelisable dynamical system with uniformly-bounded trajectories.

scholarly journals The arc-length variation of analytic capacity and a conformal geometry

1992 ◽  
Vol 125 ◽  
pp. 151-216
Author(s):  
Takafumi Murai
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Conformal Geometry ◽  
Supremum Norm ◽  
Length Variation ◽  
Analytic Capacity ◽  
Arc Length ◽  
Bounded Analytic Functions ◽  
Extended Complex

For a domain Ω in the extended complex plane C ∪{∞}, H∞(Ω) denotes the Banach space of bounded analytic functions in Ω with supremum norm ∥ · ∥H∞ For ζ ∈ Ω, we putwhere f′(∞) = lim,z→∞z{f (∞) = f(z)} if ζ = ∞.

scholarly journals Weak Sequential Convergence in Bounded Finitely Additive Measures

2020 ◽  
Vol 48 (2) ◽  
pp. 379-389
Author(s):  
Salvador López-Alfonso ◽  
Manuel López-Pellicer
Keyword(s):  
Sequential Convergence ◽  
Finitely Additive Measures ◽  
Additive Measures

scholarly journals Uniform boundedness principles for regular Borel vector measures

1980 ◽  
Vol 29 (2) ◽  
pp. 206-218 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Banach Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Uniform Boundedness ◽  
Vector Measures ◽  
Boundedness Theorem ◽  
Regular Borel Measure ◽  
Pointwise Limit ◽  
Regular Open ◽  
Uniformly Bounded

The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

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