Journal of Symbolic Logic 73 (4):1416-1432 (2008)
|Abstract||We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable separable positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA+⇁ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA+⇁ CH every atomless ccc Boolean algebra satisfies this stronger chain condition|
|Keywords||strictly positive measure chain conditions|
|Categories||categorize this paper)|
|Through your library||Configure|
Similar books and articles
Daniele Mundici (1995). Averaging the Truth-Value in Łukasiewicz Logic. Studia Logica 55 (1):113 - 127.
Bronisław Tembrowski (1983). The Theory of Boolean Algebras with an Additional Binary Operation. Studia Logica 42 (4):389 - 405.
Janusz Czelakowski (1979). Partial Boolean Algebras in a Broader Sense. Studia Logica 38 (1):1 - 16.
Nguyen Cat Ho & Helena Rasiowa (1989). Plain Semi-Post Algebras as a Poset-Based Generalization of Post Algebras and Their Representability. Studia Logica 48 (4):509 - 530.
Antoni Torrens (1987). W-Algebras Which Are Boolean Products of Members of SR and CW-Algebras. Studia Logica 46 (3):265 - 274.
Bronisław Tembrowski (1986). Q-Ultrafilters and Normal Ultrafilters in B-Algebras. Studia Logica 45 (2):167 - 179.
Roman Wencel (2003). Definable Sets in Boolean Ordered o-Minimal Structures. II. Journal of Symbolic Logic 68 (1):35-51.
Karel Prikry (1971). On Measures on Complete Boolean Algebras. Journal of Symbolic Logic 36 (3):395-406.
Hirokazu Nishimura (1991). Boolean Valued Lie Algebras. Journal of Symbolic Logic 56 (2):731-741.
Misao Nagayama (1992). On Boolean Algebras and Integrally Closed Commutative Regular Rings. Journal of Symbolic Logic 57 (4):1305-1318.
Mohamed A. Amer (1985). Extension of Relatively |Sigma-Additive Probabilities on Boolean Algebras of Logic. Journal of Symbolic Logic 50 (3):589 - 596.
J. L. Bell (1997). Zorn's Lemma and Complete Boolean Algebras in Intuitionistic Type Theories. Journal of Symbolic Logic 62 (4):1265-1279.
David Miller (2009). A Refined Geometry of Logic. Principia 13 (3):339-356.
Roberto Cignoli & Antoni Torrens Torrell (2006). Free Algebras in Varieties of Glivenko MTL-Algebras Satisfying the Equation 2(X2) = (2x). Studia Logica 83 (1-3):157 - 181.
Added to index2010-09-12
Total downloads2 ( #245,680 of 722,745 )
Recent downloads (6 months)0
How can I increase my downloads?