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On Boolean algebras with strictly positive measures [article]

Menachem Magidor, Grzegorz Plebanek
2018 arXiv   pre-print
We investigate reflection-type problems on the class SPM, of Boolean algebras carrying strictly positive finitely additive measures.  ...  We show, in particular, that in the constructible universe there is a Boolean algebra A which is not in SPM but every subalgebra of A of cardinality c admits a strictly positive measure.  ...  This research started during the conference Set-theoretic methods in topology and analysis (Bȩdlewo, September 2017) organized on the occasion of Professor Aleksander B laszczyk's 70th birthday.  ... 
arXiv:1809.04118v3 fatcat:p6s24274jvg6hccg46kf6ngrbi

Strictly positive measures on Boolean algebras

Mirna Džamonja, Grzegorz Plebanek
2008 Journal of Symbolic Logic (JSL)  
AbstractWe investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces.  ...  We show that there is aZFCexample of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure.  ...  A strictly positive measure on a Boolean algebra is a finitely or countably additive (depending on the context) measure which assigns positive value to every nonzero element of the algebra.  ... 
doi:10.2178/jsl/1230396929 fatcat:mtuilxnymvfbrfsb353o6v4c3y

Page 489 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 64, Issue 3 [page]

1948 American Mathematical Society. Transactions of the American Mathematical Society  
By restricting the domain of a strictly positive measure on a sub- algebra T of A toaset SCT (with 1 in S), we always obtain a strictly increas- ing partial measure on S.  ...  coincides with that of a strictly positive measure.  ... 

Convergence and submeasures in Boolean algebras [article]

Thomas Jech
2017 arXiv   pre-print
A Boolean algebra carries a strictly positive exhaustive submeasure if and only if it has a sequential topology that is uniformly Frechet.  ...  Let B be a complete Boolean algebra. (a) B carries a strictly positive σ−additive measure if and only if it is weakly distributive and carries a strictly positive finitely additive measure.  ...  Does the existence of a strictly positive continuous submeasure imply the existence of a strictly positive σ−additive measure?  ... 
arXiv:1705.01019v1 fatcat:at6ih336lbctxbzhg7itzzxyeu

Concerning measures on Boolean algebras

Haim Gaifman
1964 Pacific Journal of Mathematics  
The completion by cuts of the free Boolean algebra on ^o generators is also a Boolean algebra, which satisfies (*) and has no strictly positive countably additive measure (although it does have a strictly  ...  This is so since every measure induces in a natural way a strictly positive measure on the quotient algebra of the original algebra and the ideal of all elements of measure 0.  ... 
doi:10.2140/pjm.1964.14.61 fatcat:7x3ei4h76bgdzkca22xr3sivgu

Measures in Boolean algebras

Alfred Horn, Alfred Tarski
1948 Transactions of the American Mathematical Society  
The Boolean algebra in III has a strictly positive measure^2).  ...  Strictly positive measure. Definition 2.1. A measure f on a Boolean algebra A is called strictly posi- tive if fix) =0 only for x = 0.  ...  The function h just defined is easily seen to be a strictly positive measure on A which is not countably additive.  ... 
doi:10.1090/s0002-9947-1948-0028922-8 fatcat:ubpsluboxbek7ecyjscjgmyh6a

Measure recognition problem

M. Dzamonja
2006 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
MRP asks if for a given Boolean algebra and a property Φ of measures one can recognize by purely combinatorial means if supports a strictly positive measure with property Φ.  ...  The long term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem which gives  ...  One of the goals is a structure theory for Boolean algebras that support a finitely additive strictly positive measure.  ... 
doi:10.1098/rsta.2006.1896 pmid:17090453 fatcat:ibqwzg4fj5aqpdqii36fgzf6zy

On the isomorphism problem for measures on Boolean algebras

Piotr Borodulin-Nadzieja, Mirna Džamonja
2013 Journal of Mathematical Analysis and Applications  
We present an example of a Boolean algebra which supports only separable measures but no uniformly regular one. 2000 Mathematics Subject Classification. 28A60, 03E15.  ...  We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure.  ...  Suppose that µ is a non-atomic strictly positive measure on a Boolean algebra B.  ... 
doi:10.1016/j.jmaa.2013.03.053 fatcat:tdhpposfpbdbvksuetohrfk74e

On the isomorphism problem for measures on Boolean algebras [article]

Piotr Borodulin-Nadzieja, Mirna Džamonja
2011 arXiv   pre-print
We present an example of a Boolean algebra which supports only separable measures but no uniformly regular one.  ...  We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure.  ...  Suppose that µ is a non-atomic strictly positive measure on a Boolean algebra B.  ... 
arXiv:1105.1250v1 fatcat:vadw5o6mnrakvmvgoi4qxcsuuq

Measures on Boolean algebras

J. L. Kelley
1959 Pacific Journal of Mathematics  
Not all Boolean algebras possess strictly positive measures, and workable necessary and sufficient conditions for the existence of a strictly positive measure have not been given.  ...  In particular, I do not know whether there is necessarily a strictly positive measure on an algebra s?  ... 
doi:10.2140/pjm.1959.9.1165 fatcat:mlbpcovcbzffhnxaib6xxaoxe4

Preorders Compatible with Probability Measures Defined on a Boolean Algebra

M. G. Schwarze
1989 Proceedings of the American Mathematical Society  
We use nonstandard analysis techniques to find conditions for the existence of a strictly positive measure weakly compatible with a preorder defined in a Boolean algebra generated by denumerably many atoms  ...  Then there exists a strictly positive a-measure p weakly compatible with < . Counterexamples Example 1. Let B be a e-complete Boolean algebra generated by {al■: i eco}.  ...  5//, that it is enough to solve the problem for strictly positive compatible cr-measures, where a measure is strictly positive if p(a) = 0 implies a = 0.  ... 
doi:10.2307/2046962 fatcat:khyskqzgind7xc65wyeajefh54

Preorders compatible with probability measures defined on a Boolean algebra

M. G. Schwarze
1989 Proceedings of the American Mathematical Society  
We use nonstandard analysis techniques to find conditions for the existence of a strictly positive measure weakly compatible with a preorder defined in a Boolean algebra generated by denumerably many atoms  ...  Then there exists a strictly positive a-measure p weakly compatible with < . Counterexamples Example 1. Let B be a e-complete Boolean algebra generated by {al■: i eco}.  ...  5//, that it is enough to solve the problem for strictly positive compatible cr-measures, where a measure is strictly positive if p(a) = 0 implies a = 0.  ... 
doi:10.1090/s0002-9939-1989-0935109-2 fatcat:fqmpn7ux7fh4tknfq4jtfqmb7e

Fuzzy logic as a logic of the expressive strength of information

Thomas Vetterlein
2007 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
To this end, we consider a Boolean algebra endowed with an automorphism group or, alternatively, with a measure.  ...  The Boolean algebra is meant to model a collection of properties; and the additional structure is used to identify pairs of properties which, although possibly distinct, are equally strong.  ...  Definition 4. 1 1 Let B be a separable Boolean algebra, and let m be a strictly positive measure on B.  ... 
doi:10.1007/s00500-007-0208-5 fatcat:cn3lehacljawrjjgwoxnr2b5he

Category measures on Baire spaces

J. M. Ayerbe Toledano
1990 Publicacions matemàtiques  
CATEGORY MEASURES ON BAIRE SPACES J .M . AYERBE TOLEDANO The purpose of this paper is to give a necessary and sufñcient condition to define a category measure on a Baire topological space .  ...  Let A be a weakly countably distributive and countable Boolean Q-algebra.  ...  Then ¡he Stone's space X of A is in the conditions of example 3.3 and therefore it admts a category measure and ii satisfies property Proof. We argue as in proposition 3 .2. CATEGORY MEASURES  ... 
doi:10.5565/publmat_34290_09 fatcat:qidx2uu6gjakvamvb3uaisz4oi

Embeddings of metric Boolean algebras in ℝ^N [article]

Stefano Bonzio, Andrea Loi
2022 arXiv   pre-print
A Boolean algebra equipped with a (finitely-additive) positive probability measure m can be turned into a metric space ( , d_m), where d_m(a,b)= m ((a∧ b)∨( a∧ b)), for any a,b∈ A, sometimes referred to  ...  as metric Boolean algebra.  ...  Let A be a Boolean algebra equipped with a strictly positive (finitely additive) probability measure m.  ... 
arXiv:2207.14600v1 fatcat:wwktr6646nbfneaygyvlphb3u4
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