scholarly journals Finitely Additive Measures on Topological Spaces and Boolean Algebras, University of East Anglia, UK, 2015. Supervised by Mirna Džamonja

2018 ◽  
Vol 24 (2) ◽  
pp. 199-200
Author(s):  
Zanyar A. Ameen ◽  
Mirna Džamonja
Keyword(s):  
Boolean Algebras ◽  
Topological Spaces ◽  
East Anglia ◽  
Finitely Additive Measures ◽  
Additive Measures

scholarly journals Existence of nonatomic charges

1978 ◽  
Vol 25 (1) ◽  
pp. 1-6 ◽  
Author(s):  
K. P. S. Bhaskara Rao ◽  
M. Bhaskara Rao
Keyword(s):  
Boolean Algebra ◽  
Complete Characterization ◽  
Boolean Algebras ◽  
Finitely Additive Measures ◽  
Additive Measures

AbstractA complete characterization of Boolean algebras which admit nonatomic charges (i.e. finitely additive measures) is obtained. This also gives rise to a characterization of superatomic Boolean algebras. We also consider the problem of denseness of the set of all nonatomic charges in the space of all charges on a given Boolean algebra, equipped with a suitable topology.

scholarly journals Strictly positive measures on Boolean algebras

2008 ◽  
Vol 73 (4) ◽  
pp. 1416-1432 ◽  
Author(s):  
Mirna Džamonja ◽  
Grzegorz Plebanek
Keyword(s):  
Boolean Algebra ◽  
Positive Measure ◽  
Finite Measure ◽  
Chain Condition ◽  
Boolean Algebras ◽  
Radon Measures ◽  
Finitely Additive Measures ◽  
Nonatomic Measure ◽  
A Chain ◽  
Additive Measures

AbstractWe investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA + ¬ CH every atomless ccc Boolean algebra satisfies this stronger chain condition.

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
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