zbMATH — the first resource for mathematics

On mean value in F-quantum spaces. (English) Zbl 0719.60002
An F-quantum space is a family \(F\subset [0,1]^ X\) of real functions \((=\) fuzzy sets) such that (i) if \(f\in F\), then \(f^{\perp}\in F\); (ii) if \(f_ n\in F\) (n\(\geq 1)\), then \(\bigvee_{n}f_ n:=\sup_{n}f_ n\in F\). A state is a mapping m: \(F\to F\) such that (i) \(m(a\vee a^{\perp})=1\) for any \(a\in F\); (ii) m(\(\bigvee_{i}f_ i)=\sum_{i}m(f_ i)\) whenever \(f_ i+f_ j\geq 1\) for \(i\neq j\). An observable is a mapping x: B(R)\(\to F\) such that (i) \(x(E^ c)=x(E)^{\perp}\), \(E\in B(R)\); (ii) x(\(\cup_{i}E_ i)=\bigvee_{i}x(E_ i)\). Then \(m\circ x\) is a probability measure on B(R), so that \(m(x):=\int tdm\circ x\) is a mean value of x in m. For this model, an indefinite integral of observables is defined and basic properties are investigated.

60A99 Foundations of probability theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03E72 Theory of fuzzy sets, etc.
Full Text: EuDML
[1] B. Riečan: A new approach to some notions of statistical quantum mechanics. Busefal 36, 1988, 4-6.
[2] B. Riečan A. Dvurečenskij: On randomness and fuzziness. Progress in Fuzzy Sets in Europe, (Warszawa 1986), PAN, Warszawa 1988, 321-327.
[3] A. Dvurečenskij B. Riečan: On joint distribution of observables for F-quantum spaces. Fuzzy Sets and Systems. · Zbl 0814.03041
[4] A. Dvurečenskij B. Riečan: Fuzziness and commensurability. Fasciculi Mathematici. · Zbl 0731.04007
[5] A. Dvurečenskij F. Chovanec: Fuzzy quantum spaces and compatibility. Int. J. Theor. Phys. 27 (1988), 1069-1082. · Zbl 0657.60004 · doi:10.1007/BF00674352
[6] A. Tirpáková: On a sum of observables in F-quantum spaces and its applications to convergence theorems. Proc. of the First Winter School on Measure Theory (Liptovský Ján 1988), 68-76. · Zbl 0685.03044
[7] A. Dvurečenskij A. Tirpáková: A note on a sum of observables in F-quantum spaces and its properties. Busefal 36 (1988), 132-137. · Zbl 0662.03057
[8] A. Dvurečenskij A. Tirpáková: Sum of observables in fuzzy quantum soaces and convergence theorems. · Zbl 0753.60005
[9] K. Piasecki: On the extension of fuzzy P-measure generated by outer measure. Proc. 2nd Napoli Meeting on the Mathematics of Fuzzy Systems 1985, 119-135.
[10] A. Dvurečenskij: The Radon-Nikodým theorem for fuzzy probability spaces. · Zbl 0747.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.