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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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