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On Boolean algebras with strictly positive measures
[article]

2018
*
arXiv
*
pre-print

We investigate reflection-type problems

arXiv:1809.04118v3
fatcat:p6s24274jvg6hccg46kf6ngrbi
*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. ...##
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Strictly positive measures on Boolean algebras

2008
*
Journal of Symbolic Logic (JSL)
*

AbstractWe investigate

doi:10.2178/jsl/1230396929
fatcat:mtuilxnymvfbrfsb353o6v4c3y
*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*. ...##
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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*. ...##
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Convergence and submeasures in Boolean algebras
[article]

2017
*
arXiv
*
pre-print

A

arXiv:1705.01019v1
fatcat:at6ih336lbctxbzhg7itzzxyeu
*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*? ...##
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Concerning measures on Boolean algebras

1964
*
Pacific Journal of Mathematics
*

The completion by cuts of the free

doi:10.2140/pjm.1964.14.61
fatcat:7x3ei4h76bgdzkca22xr3sivgu
*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. ...##
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Measures in Boolean algebras

1948
*
Transactions of the American Mathematical Society
*

The

doi:10.1090/s0002-9947-1948-0028922-8
fatcat:ubpsluboxbek7ecyjscjgmyh6a
*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. ...##
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Measure recognition problem

2006
*
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
*

MRP asks if for a given

doi:10.1098/rsta.2006.1896
pmid:17090453
fatcat:ibqwzg4fj5aqpdqii36fgzf6zy
*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*. ...##
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On the isomorphism problem for measures on Boolean algebras

2013
*
Journal of Mathematical Analysis and Applications
*

We present an example of a

doi:10.1016/j.jmaa.2013.03.053
fatcat:tdhpposfpbdbvksuetohrfk74e
*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. ...##
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On the isomorphism problem for measures on Boolean algebras
[article]

2011
*
arXiv
*
pre-print

We present an example of a

arXiv:1105.1250v1
fatcat:vadw5o6mnrakvmvgoi4qxcsuuq
*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. ...##
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Measures on Boolean algebras

1959
*
Pacific Journal of Mathematics
*

Not all

doi:10.2140/pjm.1959.9.1165
fatcat:mlbpcovcbzffhnxaib6xxaoxe4
*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? ...##
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Preorders Compatible with Probability Measures Defined on a Boolean Algebra

1989
*
Proceedings of the American Mathematical Society
*

We use nonstandard analysis techniques to find conditions for the existence of a

doi:10.2307/2046962
fatcat:khyskqzgind7xc65wyeajefh54
*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. ...##
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Preorders compatible with probability measures defined on a Boolean algebra

1989
*
Proceedings of the American Mathematical Society
*

We use nonstandard analysis techniques to find conditions for the existence of a

doi:10.1090/s0002-9939-1989-0935109-2
fatcat:fqmpn7ux7fh4tknfq4jtfqmb7e
*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. ...##
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Fuzzy logic as a logic of the expressive strength of information

2007
*
Soft Computing - A Fusion of Foundations, Methodologies and Applications
*

To this end, we consider a

doi:10.1007/s00500-007-0208-5
fatcat:cn3lehacljawrjjgwoxnr2b5he
*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. ...##
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Category measures on Baire spaces

1990
*
Publicacions matemàtiques
*

CATEGORY

doi:10.5565/publmat_34290_09
fatcat:qidx2uu6gjakvamvb3uaisz4oi
*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*...##
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Embeddings of metric Boolean algebras in ℝ^N
[article]

2022
*
arXiv
*
pre-print

A

arXiv:2207.14600v1
fatcat:wwktr6646nbfneaygyvlphb3u4
*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. ...
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