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Boolean powers and stochastic spaces. (English) Zbl 0789.03038
We investigate the relationship between the Boolean power \(\mathbb{R}[\mathbb{B}]\) of \(\mathbb{R}\) and the elementary stochastic space \(E\) in the sense of D. Kappos [Probability algebras and stochastic spaces (1969; Zbl 0196.185)]. We obtain here that these two spaces are isomorphic. In this way, we obtain a stochastic interpretation of the Boolean power structure. The development is similar to Takeuti’s Boolean analysis. The main difference lies in the fact that we use a full Boolean-valued model, known as Boolean power, and a two-step procedure: First we develop a restrictive model (a discrete or a kind of first order model), the Boolean power in which all the axioms of the reals can be transferred immediately, and then we complete it using Cauchy sequences or Dedekind cuts in order to get a model isomorphic to the stochastic space \(V\). In this way, we avoid the general Scott-Solovay model and we get instead a model which is more appropriate for generalizing the Robinsonian Infinitesimal Analysis to Boolean Analysis.

MSC:
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
60B99 Probability theory on algebraic and topological structures
03H05 Nonstandard models in mathematics
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