Real Functions in Stochastic Dependence

Tatra Mountains Mathematical Publications

2021

Self Cite

We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the “classical core” of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.

“…into M(A), then h 0 can be uniquely extended to a sequentially continuous A-homomorphism of M(B) into M(A), cf. [1,2].…”

Section: E a ⊂ M(a) And B ⊂ M(b)mentioning

confidence: 99%

“…In the divisible probability theory, an important role is played by "degenerated" observables. They capture stochastic independence of one random experiment on another one [2,6].…”

Section: Stochastic Backgroundmentioning

confidence: 99%

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Tatra Mountains Mathematical Publications

2021

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“…Clearly, for all s ∈ P(A) we have T g (s) = q (cf. [3]). Note that a classical degenerated random variable becomes a special case (it maps all classical outcomes ω ∈ Ω to the same real number and the preimage map maps every real event either to ∅ or to Ω, and maps each probability measure on the sample events to the same Dirac measure on the real Borel measurable sets).…”

Section: Application To Stochastic Dependence/independencementioning

confidence: 99%

“…At previous conferences on Real Functions held in (Stará Lesná 2016, 2018, Ustka 2017) we have presented our results related to the transition from classical probability (cf [16]) to its fuzzification (cf. [2], [3]). The classical probability space (Ω, A, p) is extended to (Ω, M(A), (.)…”

Section: Introductionmentioning

confidence: 99%

Product of Measurable Spaces and Applications

Tatra Mountains Mathematical Publications

2019

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Divisible extension of probability

Eliáš

Papčo

2020

Mathematica Slovaca

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.