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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ \kappa $-finiteness and $ \kappa $-additivity of measures on sets and left invariant measures on discrete groups


Authors: Thomas E. Armstrong and Karel Prikry
Journal: Proc. Amer. Math. Soc. 80 (1980), 105-112
MSC: Primary 28C10; Secondary 03E55, 28A12
MathSciNet review: 574517
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Abstract: For any cardinal $ \kappa $ a possibly infinite measure $ \mu \geqslant 0$ on a set X is strongly non-$ \kappa $-additive if X is partitioned into $ \kappa $ or fewer $ \mu $-negligible sets. The measure $ \mu $ is purely non-$ \kappa $-additive if it dominates no nontrivial $ \kappa $-additive measure. The properties and relationships of these types of measures are examined in relationship to measurable ideal cardinals and real-valued measurable cardinals. Any $ \kappa $-finite left invariant measure $ \mu $ on a group G of cardinality larger than $ \kappa $ is strongly non-$ \kappa $-additive. In particular, $ \sigma $-finite left invariant measures on infinite groups are strongly finitely additive.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0574517-0
PII: S 0002-9939(1980)0574517-0
Keywords: Left invariant measures, left invariant means, $ \kappa $-additivity, $ \kappa $-finiteness, pure non-$ \kappa $-additivity, real-valued measurable cardinal, $ \kappa $-complete ideal, $ \kappa $-saturated ideal
Article copyright: © Copyright 1980 American Mathematical Society