Finitely Additive Probability

2002 ◽  
pp. 25-29
Author(s):  
Giulianella Coletti ◽  
Romano Scozzafava
Keyword(s):  
Finitely Additive Probability ◽  
Additive Probability

scholarly journals Probability logic: A model-theoretic perspective

2020 ◽  
Author(s):  
M Pourmahdian ◽  
R Zoghifard
Keyword(s):  
Characterization Theorem ◽  
Expressive Power ◽  
Probability Logic ◽  
Theoretic Analysis ◽  
Compactness Property ◽  
Probability Models ◽  
Finitely Additive Probability ◽  
Additive Probability ◽  
Basic Probability ◽  
Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

Analytic Sets, Baire Sets and the Standard Part Map

1979 ◽  
Vol 31 (3) ◽  
pp. 663-672 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Hausdorff Space ◽  
Original Problem ◽  
Nonstandard Analysis ◽  
Topological Spaces ◽  
Theoretical Structure ◽  
Standard Part ◽  
Finitely Additive Probability ◽  
The Family ◽  
Additive Probability ◽  
Analytic Sets

The problems considered here arose in connection with the interesting use by Loeb [8] and Anderson [1], [2] of Loeb's measure construction [7] to define measures on certain topological spaces. The original problem, from which the results given here developed, was to identify precisely the family of sets on which these measures are defined.To be precise, let be a set theoretical structure and * a nonstandard extension of , as in the usual framework for nonstandard analysis (see [10]). Let X be a Hausdorff space in and stx the standard part map for X, defined on the set of nearstandard points in *X. Suppose, for example, µ is an internal, finitely additive probability measure defined on the internal subsets of *X.

scholarly journals Remarks on continuously distributed sequences

2021 ◽  
Vol 13 (1) ◽  
pp. 89-97
Author(s):  
M. Paštéka
Keyword(s):  
Distribution Function ◽  
Probability Measure ◽  
Finitely Additive Probability ◽  
Additive Probability

In the first part of the paper we define the notion of the density as certain type of finitely additive probability measure and the distribution function of sequences with respect to the density. Then we derive some simple criterions providing the continuity of the distribution function of given sequence. These criterions we apply to the van der Corput's sequences. The Weyl's type criterions of continuity of the distribution function are proven.

Automorphism–invariant measures on ℵ0-categorical structures without the independence property

1996 ◽  
Vol 61 (2) ◽  
pp. 640-652
Author(s):  
Douglas E. Ensley
Keyword(s):  
Boolean Algebra ◽  
Invariant Measures ◽  
Measure Zero ◽  
Probability Measures ◽  
Independence Property ◽  
Natural Action ◽  
Zero Sets ◽  
Finitely Additive Probability ◽  
Additive Probability

AbstractWe address the classification of the possible finitely-additive probability measures on the Boolean algebra of definable subsets of M which are invariant under the natural action of Aut(M). This pursuit requires a generalization of Shelah's forking formulas [8] to “essentially measure zero” sets and an application of Myer's “rank diagram” [5] of the Boolean algebra under consideration. The classification is completed for a large class of ℵ0-categorical structures without the independence property including those which are stable.

PROBABILITY MEASURES ON PROJECTIONS IN VON NEUMANN ALGEBRAS

1989 ◽  
Vol 01 (02n03) ◽  
pp. 235-290 ◽  
Author(s):  
SHUICHIRO MAEDA
Keyword(s):  
Probability Measure ◽  
Linear Functional ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Probability Measures ◽  
Type I ◽  
Von Neumann ◽  
Positive Linear Functional ◽  
Finitely Additive Probability ◽  
Additive Probability

A state ϕ on a von Neumann algebra A is a positive linear functional on A with ϕ(1) = 1, and the restriction of ϕ to the set of projections in A is a finitely additive probability measure. Recently it was proved that if A has no type I 2 summand then every finitely additive probability measure on projections can be extended to a state on A. Here we give precise and complete arguments for proving this result.

Epistemic adjectives: Likely and probable

2017 ◽  
Author(s):  
Daniel Lassiter
Keyword(s):  
Ratio Scale ◽  
Finitely Additive Probability ◽  
Additive Probability

This chapter investigates the (near-)synonymous relative adjectives likely and probable, starting with the hypothesis that they live on an upper- and lower-bounded ratio scale. If it is correct, then the scale in question is provably equivalent to a representation in terms of finitely additive probability. This would explain the puzzle around disjunction noted in chapter 3, and it is supported by the acceptability of ratio modifiers such as three times as likely and item-by-item consideration of ratio scale axioms (with a caveat involving connectedness). The second part of the chapter turns to a theoretical puzzle: in Kennedy’s (2007) theory, likely and probable could not be relative adjectives if their scale is bounded. However, this theory is falsified on independent grounds: among other empirical problems, relative adjectives routinely occur on bounded scales. Likely and probable provide two more counter-examples to the claim that relative adjectives are restricted to open scales.

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