Divisible extension of probability
-
Roman Frič
,
Peter Eliaš
Abstract
We outline the transition from classical probability space (Ω, A, p) to its "divisible" extension, where (as proposed by L. A. Zadeh) the σ-field A of Boolean random events is extended to the class 𝓜(A) of all measurable functions into [0,1] and the σ-additive probability measure p on A is extended to the probability integral ∫(·) dp on 𝓜(A). The resulting extension of (Ω, A,p) can be described as an epireflection reflecting A to 𝓜(A) and p to ∫(·) dp.
The transition from A to 𝓜(A), resembling the transition from whole numbers to real numbers, is characterized by the extension of two-valued Boolean logic on A to multivalued Łukasiewicz logic on 𝓜(A) and the divisibility of random events: for each random event u ∈ 𝓜(A) and each positive natural number n we have u/n ∈ 𝓜(A) and ∫(u/n) dp = (1/n) ∫u dp.
From the viewpoint of category theory, objects are of the form 𝓜(A), morphisms are observables from one object into another one and serve as channels through which stochastic information is conveyed.
We study joint random experiments and asymmetrical stochastic dependence/independence of one constituent experiment on the other one. We present a canonical construction of conditional probability so that observables can be viewed as conditional probabilities.
In the present paper we utilize various published results related to "quantum and fuzzy" generalizations of the classical theory, but our ultimate goal is to stress mathematical (categorical) aspects of the transition from classical to what we call divisible probability.
(Communicated by Mirko Navara)
Acknowledgement
The authors are indebted to the referee for valuable suggestions to improve the original text.
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© 2020 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
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- The lattices of 𝔏-fuzzy state filters in state residuated lattices
- Central lifting property for orthomodular lattices
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Artikel in diesem Heft
- Regular papers
- Fuzzy deductive systems of RM algebras
- Congruence pairs of principal MS-algebras and perfect extensions
- The lattices of 𝔏-fuzzy state filters in state residuated lattices
- Central lifting property for orthomodular lattices
- EQ-Modules
- On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
- Remarks on some generalization of the notion of microscopic sets
- Disjointness of composition operators on Hv0 spaces
- A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
- The Poincaré-Cartan forms of one-dimensional variational integrals
- Coarse cohomology with twisted coefficients
- Divisible extension of probability
- Asymptotic behavior of the records of multivariate random sequences in a norm sense
- Strong convergence of the functional nonparametric relative error regression estimator under right censoring
- A new kumaraswamy generalized family of distributions: Properties and applications
- Efficient message transmission via twisted Edwards curves
- Computation of several Hessenberg determinants
