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

1997 ◽  
Vol 29 (1) ◽  
pp. 23-31
Author(s):  
Alexander G. Chentsov ◽  
O. A. Cherepanova
Keyword(s):  
Attainable Sets ◽  
Finitely Additive Measures ◽  
Additive Measures

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

scholarly journals The Foundations of Probability with Black Swans

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Graciela Chichilnisky
Keyword(s):  
Subjective Probability ◽  
Heavy Tails ◽  
Rare Events ◽  
Power Laws ◽  
Equal Treatment ◽  
Subjective Probabilities ◽  
Black Swans ◽  
Classic Theory ◽  
Finitely Additive Measures ◽  
Additive Measures

We extend the foundation of probability in samples with rare events that are potentially catastrophic, calledblack swans, such as natural hazards, market crashes, catastrophic climate change, and species extinction. Such events are generally treated as ‘‘outliers’’ and disregarded. We propose a new axiomatization of probability requiring equal treatment in the measurement of rare and frequent events—the Swan Axiom—and characterize the subjective probabilities that the axioms imply: these are neither finitely additive nor countably additive but a combination of both. They exclude countably additive probabilities as in De Groot (1970) and Arrow (1971) and are a strict subset of Savage (1954) probabilities that are finitely additive measures. Our subjective probabilities are standard distributions when the sample has no black swans. The finitely additive part assigns however more weight to rare events than do standard distributions and in that sense explains the persistent observation of ‘‘power laws’’ and ‘‘heavy tails’’ that eludes classic theory. The axioms extend earlier work by Chichilnisky (1996, 2000, 2002, 2009) to encompass the foundation of subjective probability and axiomatic treatments of subjective probability by Villegas (1964), De Groot (1963), Dubins and Savage (1965), Dubins (1975) Purves and Sudderth (1976) and of choice under uncertainty by Arrow (1971).

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