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Extension of Relatively |sigma-Additive Probabilities on Boolean Algebras of Logic

Mohamed A. Amer
The Journal of Symbolic Logic
Vol. 50, No. 3 (Sep., 1985), pp. 589-596
DOI: 10.2307/2274314
Stable URL: http://www.jstor.org/stable/2274314
Page Count: 8
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Extension of Relatively |sigma-Additive Probabilities on Boolean Algebras of Logic
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Abstract

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability on B (where B and C are Boolean algebras, and $\mathscr{B} \subseteq \mathscr{C}$) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) can be extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L.

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