Journal of Symbolic Logic

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

Mohamed A. Amer

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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 $\sigma$-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a $\mathscr{C}-\sigma$-additive probability on $\mathscr{B}$ (where $\mathscr{B}$ and $\mathscr{C}$ 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 $\bar{S}(L)-\sigma$-additive probability on $\bar{s}(L)$ can be extended (uniquely, under some conditions) to a $\sigma$-additive probability on $\bar{S}(L)$, where $L$ belongs to a quite extensive family of first order languages, and $\bar{S}(L)$ and $\bar{s}(L)$ are, respectively, the Boolean algebras of sentences and quantifier free sentences of $L$.

Article information

J. Symbolic Logic, Volume 50, Issue 3 (1985), 589-596.

First available in Project Euclid: 6 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03G05: Boolean algebras [See also 06Exx]
Secondary: 60B99: None of the above, but in this section

First order logic Boolean algebras $\sigma$-additive probabilities


Amer, Mohamed A. Extension of Relatively $|sigma$-Additive Probabilities on Boolean Algebras of Logic. J. Symbolic Logic 50 (1985), no. 3, 589--596.

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