## Relative conditional expectations on a logic.(English)Zbl 0585.60003

In this paper the notion of a conditional expectation of an observable x on a logic with respect to a sublogic $$L_ 0\subset L$$ in a state m on L, relative to an element $$a\in L$$ such that $$m(a)=1$$ and $$R(x)\subset L_ 0$$ is partially compatible with [a] is introduced. This conditional expectation is an analogue of the conditional expectation of an integrable function f on a probability space (X,S,$$\mu)$$ with respect to a sub-$$\sigma$$-field $$S_ 0$$ of S, relativized by a massive set A (i.e. $$\mu (A)=1)$$; that is, the conditional expectation of f with respect to the $$\sigma$$-field $$S_ 1$$ generated by $$S_ 0$$ and A.
For such defined conditional expectation fundamental properties are shown and conditional expectations on summable logics are analyzed. The ”relative” conditional expectation is characterized by ”measurable spaces” of observables.

### MSC:

 60A99 Foundations of probability theory 81P20 Stochastic mechanics (including stochastic electrodynamics)
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### References:

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