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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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