Common Extensions of Finitely Additive Measures and a Characterization of Cotorsion Abelian Groups

Additive Measures ◽

Positive Integers ◽

Vector Valued ◽

Increasing Functions

Let V denote a ring of subsets of an abstract space X, let R + denote the nonnegative reals, and let N denote the set of positive integers. We denote by C(V) the space of all subadditive and increasing functions, from the ring V into R +, which are zero at the empty set. The space C(V) is called the space of contents on the ring V and elements are referred to as contents. A sequence of sets An ∊ V, n ∊ N is said to be dominated if there exists a set B ∊ V such that An ⊆ B, for n = 1, 2, A content p ∊ C(V) is said to be Rickart on the ring V if lim n p(An ) = 0 for each dominated, disjoint sequence An ∊ V, n ∊ N.

Existence of nonatomic charges

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700038866 ◽

1978 ◽

Vol 25 (1) ◽

pp. 1-6 ◽

Cited By ~ 3

Author(s):

K. P. S. Bhaskara Rao ◽

M. Bhaskara Rao

Keyword(s):

Boolean Algebra ◽

Complete Characterization ◽

Boolean Algebras ◽

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.

Relaxations of Attainable Sets and Their Generalized Representation in the Class of Two-Valued Finitely Additive Measures

Journal of Automation and Information Sciences ◽

10.1615/jautomatinfscien.v29.i1.30 ◽

1997 ◽

Vol 29 (1) ◽

pp. 23-31

Author(s):

Alexander G. Chentsov ◽

O. A. Cherepanova

Keyword(s):

Attainable Sets ◽

A lattice-theoretic characterization of pure subgroups of Abelian groups

Ricerche di Matematica ◽

10.1007/s11587-021-00580-6 ◽

2021 ◽

Author(s):

M. Ferrara ◽

M. Trombetti

Keyword(s):

Abelian Group ◽

Abelian Groups ◽

Subgroup Lattice ◽

General Definition ◽

Short Paper ◽

Theoretic Characterization ◽

Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

Scaling sets and generalized scaling sets on Cantor dyadic group

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691320500198 ◽

2020 ◽

Vol 18 (04) ◽

pp. 2050019

Author(s):

Prasadini Mahapatra ◽

Divya Singh

Keyword(s):

Abelian Groups ◽

Real Case ◽

Locally Compact ◽

Wavelet Sets ◽

Locally Compact Abelian Groups ◽

Dyadic Group ◽

Compact Abelian Groups ◽

Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Spectral synthesis and residually finite-dimensional algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498817502000 ◽

2017 ◽

Vol 16 (10) ◽

pp. 1750200 ◽

Cited By ~ 1

Author(s):

László Székelyhidi ◽

Bettina Wilkens

Keyword(s):

Spectral Analysis ◽

Abelian Group ◽

Theoretical Approach ◽

Abelian Groups ◽

Spectral Synthesis ◽

Residually Finite ◽

Finite Dimensional ◽

Analysis And Synthesis ◽

Spectral Analysis And Synthesis

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171293000031 ◽

1993 ◽

Vol 16 (1) ◽

pp. 33-40 ◽

Cited By ~ 1

Author(s):

Robert W. Shutz

Keyword(s):

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

Forum Mathematicum ◽

10.1515/forum-2020-0317 ◽

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 ◽

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.