## Journal of Symbolic Logic

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

Mohamed A. Amer

#### 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 $\sigma$-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

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183741897

Mathematical Reviews number (MathSciNet)
MR805669

Zentralblatt MATH identifier
0581.03043

JSTOR
Amer, Mohamed A. Extension of Relatively $|sigma$-Additive Probabilities on Boolean Algebras of Logic. J. Symbolic Logic 50 (1985), no. 3, 589--596. https://projecteuclid.org/euclid.jsl/1183741897