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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.

MSC:
60A99 Foundations of probability theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03E72 Theory of fuzzy sets, etc.
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References:
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