Inequalities for λ-additive measures based on the application of the general Poincaré formula for λ-additive measures

2020 ◽  
Vol 396 ◽  
pp. 152-162 ◽  
Author(s):  
József Dombi ◽  
Tamás Jónás
Keyword(s):  
Additive Measures ◽  
Poincaré Formula

scholarly journals An Elementary Proof of the General Poincaré Formula for λ-additive Measures

2019 ◽  
Vol 24 (2) ◽  
pp. 173-185 ◽  
Author(s):  
József Dombi ◽  
Tamás Jónás
Keyword(s):  
Probability Theory ◽  
General Formula ◽  
Elementary Proof ◽  
Additive Measure ◽  
Limit Case ◽  
Additive Measures ◽  
Poincaré Formula

In a previous paper of ours [4], we presented the general formula for lambda-additive measure of union of n sets and gave a proof of it. That proof is based on the fact that the lambda-additive measure is representable. In this study, a novel and elementary proof of the formula for lambda-additive measure of the union of n sets is presented. Here, it is also demonstrated that, using elementary techniques, the well-known Poincare formula of probability theory is just a limit case of our general formula.

The general Poincaré formula for λ-additive measures

2019 ◽  
Vol 490 ◽  
pp. 285-291 ◽  
Author(s):  
József Dombi ◽  
Tamás Jónás
Keyword(s):  
Additive Measures ◽  
Poincaré Formula

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.

A note on the tau-additive measures

2021 ◽  
Author(s):  
Tingsu Yan ◽  
Yao Ouyang ◽  
Jun Li
Keyword(s):  
Additive Measures

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 THE INTERVAL-VALUED INTUITIONISTIC FUZZY GEOMETRIC CHOQUET AGGREGATION OPERATOR BASED ON THE GENERALIZED BANZHAF INDEX AND 2-ADDITIVE MEASURE

2015 ◽  
Vol 21 (2) ◽  
pp. 186-215 ◽  
Author(s):  
Fanyong MENG ◽  
Qiang ZHANG ◽  
Jiaquan ZHAN
Keyword(s):  
Geometric Mean ◽  
Fuzzy Measure ◽  
Entropy Measure ◽  
Additive Measure ◽  
Banzhaf Index ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Environment ◽  
Operational Laws ◽  
Additive Measures ◽  
Interval Valued

Based on the operational laws on interval-valued intuitionistic fuzzy sets, the generalized Banzhaf interval-valued intuitionistic fuzzy geometric Choquet (GBIVIFGC) operator is proposed, which is also an interval-valued intuitionistic fuzzy value. It is worth pointing out that the GBIVIFGC operator can be seen as an extension of some geometric mean operators. Since the fuzzy measure is defined on the power set, it makes the problem exponentially complex. In order to overall reflect the interaction among elements and reduce the complexity of solving a fuzzy measure, we further introduce the GBIVIFGC operator w.r.t. 2-additive measures. Furthermore, if the information about weights of experts and attributes is incompletely known, the models of obtaining the optimal 2-additive measures on criteria set and expert set are given by using the introduced cross entropy measure and the Banzhaf index. Finally, an approach to pattern recognition and multi-criteria group decision making under interval-valued intuitionistic fuzzy environment is developed, respectively.

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