scholarly journals Topological Aspects of the Product of Lattices

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Carmen Vlad
Keyword(s):  
Detailed Analysis ◽  
Finitely Additive Measures ◽  
Additive Measures ◽  
Nontrivial Zero

Let be an arbitrary nonempty set and a lattice of subsets of such that , . () denotes the algebra generated by , and () denotes those nonnegative, finite, finitely additive measures on (). In addition, () denotes the subset of () which consists of the nontrivial zero-one valued measures. The paper gives detailed analysis of products of lattices, their associated Wallman spaces, and products of a variety of measures.

scholarly journals Normal characterizations of lattices

2001 ◽  
Vol 28 (10) ◽  
pp. 561-570
Author(s):  
Carmen D. Vlad
Keyword(s):  
Topological Properties ◽  
Finitely Additive Measures ◽  
Additive Measures ◽  
Nontrivial Zero

LetXbe an arbitrary nonempty set andℒa lattice of subsets ofXsuch that∅,X∈ℒ. Let𝒜(ℒ)denote the algebra generated byℒandI(ℒ)denote those nontrivial, zero-one valued, finitely additive measures on𝒜(ℒ). In this paper, we discuss some of the normal characterizations of lattices in terms of the associated lattice regular measures, filters and outer measures. We consider the interplay between normal lattices, regularity orσ-smoothness properties of measures, lattice topological properties and filter correspondence. Finally, we start a study of slightly, mildly and strongly normal lattices and express then some of these results in terms of the generalized Wallman spaces.

scholarly journals Remarks onμ″-measurbale sets: regularity,σ-smootheness, and measurability

1999 ◽  
Vol 22 (2) ◽  
pp. 391-400
Author(s):  
Carman Vlad
Keyword(s):  
Regularity Conditions ◽  
Finitely Additive Measures ◽  
Additive Measures ◽  
Nontrivial Zero

LetXbe an arbitrary nonempty set andℒa lattice of subsets ofXsuch thatϕ,X∈ℒ.𝒜(ℒ)is the algebra generated byℒandℳ(ℒ)denotes those nonnegative, finite, finitely additive measuresμon𝒜(ℒ).I(ℒ)denotes the subset ofℳ(ℒ)of nontrivial zero-one valued measures. Associated withμ∈I(ℒ)(orIσ(ℒ)) are the outer measuresμ′andμ″considered in detail. In addition, measurability conditions and regularity conditions are investigated and specific characteristics are given for𝒮μ″, the set ofμ″-measurable sets. Notions of stronglyσ-smooth and vaguely regular measures are also discussed. Relationships between regularity,σ-smoothness and measurability are investigated for zero-one valued measures and certain results are extended to the case of a pair of latticesℒ1,ℒ2whereℒ1⊂ℒ2.

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.

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 Completeness and interpolation of almost-everywhere quantification over finitely additive measures

2013 ◽  
Vol 59 (4-5) ◽  
pp. 286-302
Author(s):  
João Rasga ◽  
Wafik Boulos Lotfallah ◽  
Cristina Sernadas
Keyword(s):  
Almost Everywhere ◽  
Finitely Additive Measures ◽  
Additive Measures
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